 | Lê Trung Kiên 10 điểm |
 | Đinh Trung Đức 9A 10 điểm |
 | Đinh Trung Đức 10 điểm |
 | Đinh Trung Đức 10 điểm |
 | Nguyen Tan Phuc 10 điểm |
Có 509 người đã làm bài
Cho hình vẽ. Biết \(\widehat{@p.t0@@p.t4@@p.t3@}[email protected]@^o\) , \(\stackrel\frown{@p.t2@[email protected]@}[email protected]@^o\). Tính \(\stackrel\frown{@p.t0@[email protected]@}.\)
Hoàn thành bài giải:
Do góc @p.t0@@p.t4@@p.t3@ là góc có đỉnh ở trong đường tròn || có đỉnh ở ngoài đường tròn nên \(\widehat{@p.t3@@p.t4@@p.t0@}=\dfrac{\stackrel\frown{@p.t0@[email protected]@}+\stackrel\frown{@p.t2@[email protected]@}}{2}\) || \(\widehat{@p.t3@@p.t4@@p.t0@}=\dfrac{\stackrel\frown{@p.t0@[email protected]@}-\stackrel\frown{@p.t2@[email protected]@}}{2}\) .
\(\Rightarrow\)\(\widehat{@p.t0@[email protected]@}=\) \(2.\widehat{@p.t3@@p.t4@@p.t0@}-\widehat{@p.t2@[email protected]@}\).
\(\widehat{@p.t0@[email protected]@}=\) \(@p.da@\) || \(@p.da1@\) || \(@p.da2@\) o.
p.co = shuffle(['green', 'blue', 'orange', 'purple']);
p.t = ["A","D","B","E","G","H","I","K","L","M","N","P","C","Q","F"];
p.s = randomArray(5,0,14);
p.b = random(40,45);
p.a = random(p.b-2, p.b-5);
p.nh = shuffle([rand(3,-3,3,[0]), rand(3,1,4,[]), rand(3,-1,-4,[])])[0];
params({nh: p.nh, co: p.co,s: p.s, b: p.b, a: p.a});
p.t0 = p.t[p.s[0]];
p.t1 = p.t[p.s[1]];
p.t2 = p.t[p.s[2]];
p.t3 = p.t[p.s[3]];
p.t4 = p.t[p.s[4]];
p.da = p.a*2-p.b;
p.da1 = p.a*2+p.b;
p.da2 = p.a*2-p.b+5*p.nh[0];
p.event = function(Zone){
Zone.find('.svgedit line').attr({"stroke-width": "2", "stroke": p.co[2]});
Zone.find('.svgedit path').attr({"stroke-width": "2", "stroke": p.co[3]});
Zone.find('.svgedit [rx = "93"]').attr({"stroke-width": "2", "stroke": p.co[0]});
};
p.mathFont = 0;Cho @p.t0@@p.t1@@p.t2@@p.t3@ là tứ giác nội tiếp. Kéo thả các câu trả lời thích hợp vào chỗ trống:
\(\widehat{@p.t0@@p.t1@@p.t2@}+\) \(\widehat{@p.t0@@p.t3@@p.t2@}\) || \(\widehat{@p.t1@@p.t2@@p.t3@}\) = \(180^o\)
\(\widehat{@p.t1@@p.t0@@p.t2@}=\) \(\widehat{@p.t1@@p.t3@@p.t2@}\) || \(\widehat{@p.t0@@p.t2@@p.t1@}\) .
p.t = ["A","D","B","E","G","H","I","K","L","M","N","P","C","Q","F"];
p.s = randomArray(4,0,14);
params({s: p.s});
p.t0 = p.t[p.s[0]];
p.t1 = p.t[p.s[1]];
p.t2 = p.t[p.s[2]];
p.t3 = p.t[p.s[3]];
Cho hai đoạn thẳng @p.t0@@p.t1@ và @p.t2@@p.t3@ cắt nhau tại @p.t4@. Biết rằng @p.t0@@p.t4@[email protected]@@p.t1@ = @p.t2@@p.t4@[email protected]@@p.t3@. Chứng minh rằng tứ giác @p.t0@@p.t2@@p.t1@@p.t3@ là tứ giác nội tiếp.
Điền kí hiệu thích hợp vào ô trống để được lời giải hoàn chỉnh.
|
Ta có : @p.t0@@p.t4@[email protected]@@p.t1@ = @p.t2@@p.t4@[email protected]@@p.t3@ \(\Rightarrow\) \(\dfrac{@p.t0@@p.t4@}{@p.t2@@p.t4@}=\dfrac{@p.t4@@p.t3@}{@p.t4@@p.t1@}\) || \(\dfrac{@p.t0@@p.t4@}{@p.t2@@p.t4@}=\dfrac{@p.t4@@p.t2@}{@p.t4@@p.t3@}\) . Xét \(\Delta @p.t0@@p.t4@@p.t3@\) và \(\Delta @p.t2@@p.t4@@p.t1@\) có: Góc \(\widehat{@p.t4@_1}=\widehat{@p.t4@_2}\) (Hai góc đối đỉnh) \(\dfrac{@p.t0@@p.t4@}{@p.t2@@p.t4@}=\dfrac{@p.t4@@p.t3@}{@p.t4@@p.t2@}\) (cmt) \(\Rightarrow\Delta @p.t0@@p.t4@@p.t3@\sim\Delta @p.t2@@p.t4@@p.t1@\left(c-g-c\right)\) \(\Rightarrow\) \(\widehat{@p.t2@@p.t1@@p.t0@}=\widehat{@p.t2@@p.t3@@p.t0@}\) || \(\widehat{@p.t2@@p.t1@@p.t0@}=\widehat{@p.t4@@p.t0@@p.t3@}\) . Hai \(\widehat{@p.t2@@p.t1@@p.t0@},\widehat{@p.t2@@p.t3@@p.t0@}\) cùng nhìn đoạn \(@p.t2@@p.t0@\) || \(@p.t0@@p.t3@\) dưới một góc bằng nhau nên tứ giác @p.t0@@p.t2@@p.t1@@p.t3@ là tứ giác nội tiếp. |
p.co = shuffle(['green', 'blue', 'orange', 'purple', 'brown', 'BlueViolet', 'CornflowerBlue', 'DarkCyan', 'MediumSeaGreen ']);
p.t = ["A","D","B","E","G","H","I","K","L","M","N","P","C","Q","F"];
p.s = randomArray(5,0,14);
params({co: p.co,s: p.s});
p.t0 = p.t[p.s[0]];
p.t1 = p.t[p.s[1]];
p.t2 = p.t[p.s[2]];
p.t3 = p.t[p.s[3]];
p.t4 = p.t[p.s[4]];
p.event = function(Zone){
Zone.find('.svgedit line').attr({"stroke-width": "2", "stroke": p.co[2]});
Zone.find('.svgedit path').attr({"stroke-width": "2", "stroke": p.co[3]});
Zone.find('.svgedit [rx = "107"]').attr({"stroke-width": "2", "stroke": p.co[0]});
Zone.find('.mathdefault, .katex, .katexdisplay').css({'font-family': 'Segoe UI', 'font-size': '1em', 'font-style': 'normal'});
};
Giải phương trình: [email protected](p.pt).tex()@ = @p.vp.tex()@$
Trả lời: $x = $
Điều kiện: $x \ne @p.x1@$ và $x \ne @p.dau[p.t[2]]*p.n[2]@$.
[email protected](p.pt).tex()@ = @p.vp.tex()@ ⇒ @p.ts.tex()@ = @p.x3.tex()@$
$⇔ @p.m.rutgon().dsort().tex()@ = 0 ⇔ @p.m.tex()@ = 0$. Suy ra $x= @p.x2@$ hoặc $x = @p.x1@$ (loại).
Vậy $x = @p.x2@$.
require("btds");
require("mathtype");
p.toolbar = ["sqrt","frac"];
p.n = randomArray(3,1,7);
p.dau = [-1,1]
p.t = [random(0,1), random(0,1),random(0,1)];
params({n: p.n, t: p.t});
p.x1 = p.dau[p.t[0]]*p.n[0];
p.x2 = p.dau[p.t[1]]*p.n[1];
p.x3 = new btds("(x-" + p.dau[p.t[2]]*p.n[2] + ")");
p.m = new btds("(x-" + p.x1 + ")(x-" + p.x2 + ")");
p.ts = p.m.rutgon().cong(p.x3).rutgon().dsort();
p.pt = new btds("((x-" + (p.dau[p.t[2]]*p.n[2]) + ")(x-" + p.x1 + "))");
p.vp = new btds("\\dfrac{1}{x-" +p.x1 + "}");p.a = [rand(1,-90,90,[]), rand(1,-45,45,[]), rand(1,-45,45,[]), rand(1,-45,45,[]), rand(1,-90,90,[]), rand(1,-90,90,[])];
p.g = random(9,16);
p.ko = [
'<svg width="131" height="131" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[0]+ ' 65.30556488037108,65.30558013916016) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="65.30558" cx="65.30557"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="65.06982" cx="64.8056"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m18.56374,110.24291c0,0 33.33317,-106.66614 33.33317,-106.66614c0,0 72.72691,85.80963 72.72691,85.80963"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" fill-opacity="null" fill="none" d="m45.83633,22.36455c5.45452,4.2424 18.18173,-4.2424 18.18173,-4.2424"/> <text y="87.81878" xml:space="preserve" x="57.95749" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> </g> </svg>',
'<svg width="221" height="194" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[1] + ' 110.41763305664,63.94554138183604) " id="svg_2"> <ellipse transform="rotate(5 99.32904815673783,117.21231079101555) " stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="117.21232" cx="99.32905"/> <ellipse transform="rotate(5 98.85153198242188,116.93387603759767) " stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="116.93388" cx="98.85153"/> <path transform="rotate(-117 119.8472900390625,12.205287933349604) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m66.81726,65.53836c0,0 33.33317,-106.66614 33.33317,-106.66614c0,0 72.72691,85.80963 72.72691,85.80963"/> <text y="139.84319" xml:space="preserve" x="90.72475" transform="rotate(5 100.0468749999994,131.64843749999997) " text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <path transform="rotate(5 93.93338012695321,43.43129348754887) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_4" fill-opacity="null" fill="none" d="m86.93338,37.89152c9,-5 14,12 14,12"/> </g> </svg>',
'<svg width="217" height="177.00000000000003" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[2] + '105.30556488037111,95.43053436279298) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="106.30558" cx="105.30557"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="105.06982" cx="104.8056"/> <text y="127.81878" xml:space="preserve" x="97.95749" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_4" fill-opacity="null" fill="none"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m54.86385,145.7185c0,0 27.27261,-100.60562 26.95397,-100.87021c0.31864,0.26459 -37.25696,-24.58379 -37.57559,-24.84838"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" fill-opacity="null" fill="none" d="m71.22742,37.23412c-6.06058,4.84846 7.2727,21.21203 7.2727,21.21203"/> </g> </svg>',
'<svg width="196" height="200" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[3]+ ' 108.1249923706054,98.43058013916017) " id="svg_2"> <ellipse cx="95.30557" cy="115.30558" fill="none" id="svg_1" rx="64.55558" ry="64.55558" stroke="#000" stroke-width="1.5"/> <ellipse cx="94.8056" cy="115.06982" fill="#000000" id="svg_3" rx="2" ry="2" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <text fill="#000000" fill-opacity="null" font-family="Helvetica, Arial, sans-serif" font-size="24" id="svg_7" stroke="#000" stroke-opacity="null" stroke-width="0" text-anchor="start" x="87.95749" xml:space="preserve" y="137.81878">O</text> <path d="m185.5,123.2358l-87.5,-106.2358c0.5,0.2358 55.5,124.2358 55.5,124.2358" fill="none" fill-opacity="null" id="svg_4" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <line fill="none" fill-opacity="null" id="svg_5" stroke="#000" stroke-dasharray="5,5" stroke-linecap="null" stroke-linejoin="null" stroke-opacity="null" stroke-width="1.5" x1="94.5" x2="145.5" y1="113.90246" y2="72.90246"/> <rect fill="none" fill-opacity="null" height="8" id="svg_6" stroke="#000" stroke-opacity="null" stroke-width="1.5" transform="rotate(48 144.49999999999997,79.90248107910158) " width="10" x="139.5" y="75.90248"/> <path d="m108.33337,41.49038c4.66668,4.66668 7.33335,-3.33334 7.33335,-3.33334" fill="none" fill-opacity="null" id="svg_8" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> </g> </svg>'
];
p.co = [
'<svg width="190" height="200" xmlns="http://www.w3.org/2000/svg"> <g id="svg_2" transform="rotate('+p.a[4]+' 97.57999420166037,82.55534362792973) "> <ellipse cx="81.97228" cy="89.66464" fill="none" id="svg_1" rx="64.55558" ry="64.55558" stroke="#000" stroke-width="1.5"/> <ellipse cx="81.47231" cy="89.42888" fill="#000000" id="svg_3" rx="2" ry="2" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <text fill="#000000" fill-opacity="null" font-family="Helvetica, Arial, sans-serif" font-size="24" id="svg_7" stroke="#000" stroke-opacity="null" stroke-width="0" text-anchor="start" x="47.95763" xml:space="preserve" y="102.94711">O</text> <line fill="none" fill-opacity="null" id="svg_5" stroke="#000" stroke-dasharray="5,5" stroke-linecap="null" stroke-linejoin="null" stroke-opacity="null" stroke-width="1.5" x1="81.16671" x2="132.16671" y1="88.26152" y2="47.26152"/> <rect fill="none" fill-opacity="null" height="8" id="svg_6" stroke="#000" stroke-opacity="null" stroke-width="1.5" transform="rotate(48 125.52571105957031,46.56926345825196) " width="10" x="120.52571" y="42.56926"/> <line fill="none" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" x1="101.84612" y1="10.89047" x2="177.74329" y2="103.71065" id="svg_4" stroke-linejoin="null" stroke-linecap="null" stroke="#000"/> <line fill="none" stroke="#000" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" x1="131.07679" y1="46.78777" x2="141.84598" y2="113.45421" id="svg_9" stroke-linejoin="null" stroke-linecap="null"/> <path fill="none" stroke="#000" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" d="m133.64089,67.81334c4.61537,4.10255 12.82047,-3.07691 12.51245,-3.19819" id="svg_10"/> </g> </svg>',
'<svg width="185" height="173" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[5]+ ' 97.57999420165957,82.55534362792964) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="89.66464" cx="81.97228"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="89.42888" cx="81.47231"/> <text y="102.94711" xml:space="preserve" x="47.95763" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <line y2="47.26152" y1="88.26152" x2="132.16671" x1="81.16671" stroke-width="1.5" stroke-opacity="null" stroke-linejoin="null" stroke-linecap="null" stroke-dasharray="5,5" stroke="#000" id="svg_5" fill-opacity="null" fill="none"/> <rect y="50.34701" x="126.6368" width="10" transform="rotate(51 131.63679504394528,54.34701538085939) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" height="8" fill-opacity="null" fill="none"/> <line stroke="#000" stroke-linecap="null" stroke-linejoin="null" id="svg_4" y2="103.71065" x2="177.74329" y1="10.89047" x1="101.84612" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> <line stroke="#000" stroke-linecap="null" stroke-linejoin="null" id="svg_9" y2="153.45408" x2="89.62391" y1="46.78777" x1="131.07683" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> <path stroke="#000" transform="rotate(-8 121.03442382812507,52.36984634399424) " id="svg_8" d="m125.40613,36.27217c-15.47044,8.66799 -5.69964,32.19535 -5.69964,32.19535" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> </g> </svg>'
];
params({g: p.g, a: p.a});|
Cho biết sđ \(\stackrel\frown{\text{AmB}}=\) @4*p.g@o. \(\widehat{\text{BAx}}\) = o. |
p.a = [rand(1,-90,90,[]), rand(1,-45,45,[]), rand(1,-45,45,[]), rand(1,-45,45,[]), rand(1,-90,90,[]), rand(1,-90,90,[])];
p.g = random(9,16);
params({a: p.a, g: p.g});
p.ko = [
'<svg width="131" height="131" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[0]+ ' 65.30556488037108,65.30558013916016) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="65.30558" cx="65.30557"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="65.06982" cx="64.8056"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m18.56374,110.24291c0,0 33.33317,-106.66614 33.33317,-106.66614c0,0 72.72691,85.80963 72.72691,85.80963"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" fill-opacity="null" fill="none" d="m45.83633,22.36455c5.45452,4.2424 18.18173,-4.2424 18.18173,-4.2424"/> <text y="87.81878" xml:space="preserve" x="57.95749" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> </g> </svg>',
'<svg width="221" height="194" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[1] + ' 110.41763305664,63.94554138183604) " id="svg_2"> <ellipse transform="rotate(5 99.32904815673783,117.21231079101555) " stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="117.21232" cx="99.32905"/> <ellipse transform="rotate(5 98.85153198242188,116.93387603759767) " stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="116.93388" cx="98.85153"/> <path transform="rotate(-117 119.8472900390625,12.205287933349604) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m66.81726,65.53836c0,0 33.33317,-106.66614 33.33317,-106.66614c0,0 72.72691,85.80963 72.72691,85.80963"/> <text y="139.84319" xml:space="preserve" x="90.72475" transform="rotate(5 100.0468749999994,131.64843749999997) " text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <path transform="rotate(5 93.93338012695321,43.43129348754887) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_4" fill-opacity="null" fill="none" d="m86.93338,37.89152c9,-5 14,12 14,12"/> </g> </svg>',
'<svg width="217" height="177.00000000000003" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[2] + '105.30556488037111,95.43053436279298) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="106.30558" cx="105.30557"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="105.06982" cx="104.8056"/> <text y="127.81878" xml:space="preserve" x="97.95749" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_4" fill-opacity="null" fill="none"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m54.86385,145.7185c0,0 27.27261,-100.60562 26.95397,-100.87021c0.31864,0.26459 -37.25696,-24.58379 -37.57559,-24.84838"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" fill-opacity="null" fill="none" d="m71.22742,37.23412c-6.06058,4.84846 7.2727,21.21203 7.2727,21.21203"/> </g> </svg>',
'<svg width="196" height="200" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[3]+ ' 108.1249923706054,98.43058013916017) " id="svg_2"> <ellipse cx="95.30557" cy="115.30558" fill="none" id="svg_1" rx="64.55558" ry="64.55558" stroke="#000" stroke-width="1.5"/> <ellipse cx="94.8056" cy="115.06982" fill="#000000" id="svg_3" rx="2" ry="2" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <text fill="#000000" fill-opacity="null" font-family="Helvetica, Arial, sans-serif" font-size="24" id="svg_7" stroke="#000" stroke-opacity="null" stroke-width="0" text-anchor="start" x="87.95749" xml:space="preserve" y="137.81878">O</text> <path d="m185.5,123.2358l-87.5,-106.2358c0.5,0.2358 55.5,124.2358 55.5,124.2358" fill="none" fill-opacity="null" id="svg_4" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <line fill="none" fill-opacity="null" id="svg_5" stroke="#000" stroke-dasharray="5,5" stroke-linecap="null" stroke-linejoin="null" stroke-opacity="null" stroke-width="1.5" x1="94.5" x2="145.5" y1="113.90246" y2="72.90246"/> <rect fill="none" fill-opacity="null" height="8" id="svg_6" stroke="#000" stroke-opacity="null" stroke-width="1.5" transform="rotate(48 144.49999999999997,79.90248107910158) " width="10" x="139.5" y="75.90248"/> <path d="m108.33337,41.49038c4.66668,4.66668 7.33335,-3.33334 7.33335,-3.33334" fill="none" fill-opacity="null" id="svg_8" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> </g> </svg>'
];
p.co = [
'<svg width="190" height="200" xmlns="http://www.w3.org/2000/svg"> <g id="svg_2" transform="rotate('+p.a[4]+' 97.57999420166037,82.55534362792973) "> <ellipse cx="81.97228" cy="89.66464" fill="none" id="svg_1" rx="64.55558" ry="64.55558" stroke="#000" stroke-width="1.5"/> <ellipse cx="81.47231" cy="89.42888" fill="#000000" id="svg_3" rx="2" ry="2" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <text fill="#000000" fill-opacity="null" font-family="Helvetica, Arial, sans-serif" font-size="24" id="svg_7" stroke="#000" stroke-opacity="null" stroke-width="0" text-anchor="start" x="47.95763" xml:space="preserve" y="102.94711">O</text> <line fill="none" fill-opacity="null" id="svg_5" stroke="#000" stroke-dasharray="5,5" stroke-linecap="null" stroke-linejoin="null" stroke-opacity="null" stroke-width="1.5" x1="81.16671" x2="132.16671" y1="88.26152" y2="47.26152"/> <rect fill="none" fill-opacity="null" height="8" id="svg_6" stroke="#000" stroke-opacity="null" stroke-width="1.5" transform="rotate(48 125.52571105957031,46.56926345825196) " width="10" x="120.52571" y="42.56926"/> <line fill="none" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" x1="101.84612" y1="10.89047" x2="177.74329" y2="103.71065" id="svg_4" stroke-linejoin="null" stroke-linecap="null" stroke="#000"/> <line fill="none" stroke="#000" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" x1="131.07679" y1="46.78777" x2="141.84598" y2="113.45421" id="svg_9" stroke-linejoin="null" stroke-linecap="null"/> <path fill="none" stroke="#000" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" d="m133.64089,67.81334c4.61537,4.10255 12.82047,-3.07691 12.51245,-3.19819" id="svg_10"/> </g> </svg>',
'<svg width="185" height="173" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[5]+ ' 97.57999420165957,82.55534362792964) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="89.66464" cx="81.97228"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="89.42888" cx="81.47231"/> <text y="102.94711" xml:space="preserve" x="47.95763" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <line y2="47.26152" y1="88.26152" x2="132.16671" x1="81.16671" stroke-width="1.5" stroke-opacity="null" stroke-linejoin="null" stroke-linecap="null" stroke-dasharray="5,5" stroke="#000" id="svg_5" fill-opacity="null" fill="none"/> <rect y="50.34701" x="126.6368" width="10" transform="rotate(51 131.63679504394528,54.34701538085939) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" height="8" fill-opacity="null" fill="none"/> <line stroke="#000" stroke-linecap="null" stroke-linejoin="null" id="svg_4" y2="103.71065" x2="177.74329" y1="10.89047" x1="101.84612" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> <line stroke="#000" stroke-linecap="null" stroke-linejoin="null" id="svg_9" y2="153.45408" x2="89.62391" y1="46.78777" x1="131.07683" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> <path stroke="#000" transform="rotate(-8 121.03442382812507,52.36984634399424) " id="svg_8" d="m125.40613,36.27217c-15.47044,8.66799 -5.69964,32.19535 -5.69964,32.19535" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> </g> </svg>'
];|
Cho biết \(\widehat{\text{BAy}}\) = @180 - 2*p.g@o, sđ \(\stackrel\frown{\text{AnB}}\) bằng:
|
require("btds");
require("mathtype");
p.n = rand(2,-1,-10,[0]);
p.m = rand(2,1,2,[0]);
params({n: p.n, m: p.m});
p.x1 = new btds((p.n[0]) + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
if(equalEq(p.x1.tex(), p.x2.tex())) p.n[0]--;
p.x1 = new btds(p.n[0] + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
p.btcong = p.x1.cong(p.x2);
p.btnhan = p.x1.nhan(p.x2);
p.pt = new btds("(x-" + (p.x1.rutgon().tex()) + ")(x-" + (p.x2.rutgon().tex()) + ")");
Phương trình [email protected]().tex()@ = 0$ có hai nghiệm $x_1$ và $x_2$.
Tính: $x_1^2 + x_2^2=$ .
Theo định lí Vi-ét ta có: [email protected](p.x2).rutgon().tex()@$ và [email protected](p.x2).rutgon().tex()@$
$x_1^2 + x_2^2=(x_1+x_2)^2-2x_1 x_2 = (@p.btcong.rutgon().tex()@)^2 - [email protected]().tex()@[email protected](2).tru(p.btnhan.cong(p.btnhan)).rutgon().tex()@$.
p.gcd = function(a,b) {
if (b) {
return p.gcd(b,a % b);
} else {
return Math.abs(a);
}
};
require("btds");
require("mathtype");
p.n = rand(2,-1,-4,[0]);
p.m = rand(2,1,3,[0]);
params({n: p.n, m: p.m});
p.x1 = new btds((p.n[0]) + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
if(equalEq(p.x1.tex(), p.x2.tex())) p.n[0]--;
p.x1 = new btds(p.n[0] + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
p.btcong = p.x1.cong(p.x2);
p.btnhan = p.x1.nhan(p.x2);
p.kq = new btds("1/" + p.x1.pow(2).tex() + "+ 1/" + p.x2.pow(2).tex());
p.pt = new btds((p.m[0]*p.m[1]) + "*(x-" + (p.x1.rutgon().tex()) + ")(x-" + (p.x2.rutgon().tex()) + ")");
p.hint = {
title : "Gợi ý",
content : '<span class="math-q">\\dfrac{1}{x_1^2} + \\dfrac{1}{x_2^2}=\\dfrac{(x_1+x_2)^2 - 2x_1 x_2}{(x_1 . x_2)^2} </span>',
time: 10
};Phương trình [email protected]().tex()@ = 0$ có hai nghiệm $x_1$ và $x_2$.
Tính: $\dfrac{1}{x_1^2} + \dfrac{1}{x_2^2}=$ .
Theo định lí Vi-ét ta có: [email protected](p.x2).rutgon().tex()@$ và [email protected](p.x2).rutgon().tex()@$
$\dfrac{1}{x_1^2} + \dfrac{1}{x_2^2}=\dfrac{(x_1+x_2)^2 - 2x_1 x_2}{(x_1 . x_2)^2} = \dfrac{(@p.btcong.rutgon().tex()@)^2 - [email protected]().tex()@}{@p.btnhan.rutgon().pow(2).tex()@}[email protected]().tex()@$.
Tìm hai số biết rằng tổng của chúng bằng [email protected](p.x2).rutgon().tex()@$ và tích của chúng bằng [email protected](p.x2).rutgon().tex()@$.
Trả lời: hai số đó là và .
Hai số cần tìm là nghiệm của phương trình: [email protected]().rutgon().tex()@=0$.
Giải phương trình ta nhận được hai nghiệm [email protected]().tex()@$ và [email protected]().tex()@$.
require("btds");
require("mathtype");
p.n = rand(2,-7,7,[0]);
p.m = rand(2,-3,3,[0]);
params({n: p.n, m: p.m});
p.x1 = new btds(p.n[0] + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
p.pt = new btds("(x-" + (p.x1.rutgon().tex()) + ")(x-" + (p.x2.rutgon().tex()) + ")");
p.check = function(){
var ans = getEq(Zone);
var result = 2;
if((equalEq(ans[0], p.x1.tex()) && equalEq(ans[1], p.x2.tex())) || (equalEq(ans[0], p.x2.tex()) && equalEq(ans[1], p.x1.tex()))) result = 1;
else result = 0;
return {answer: ans, result: result};
};
Cho hình bình hành @p.t0@@p.t1@@p.t2@@p.t3@. Gọi O là tâm đường tròn ngoại tiếp tam giác @p.t0@@p.t3@@p.t2@. Đường tròn (O) cắt đường thẳng @p.t2@@p.t1@ tại @p.t4@ khác @p.t2@. Chứng minh rằng @p.t0@@p.t1@ = @p.t0@@p.t4@.
Sắp xếp các dòng sau theo thứ tự thích hợp để hoàn thành bài giải.
p.co = shuffle(['green', 'blue', 'orange', 'purple', 'brown', 'BlueViolet', 'CornflowerBlue', 'DarkCyan', 'MediumSeaGreen ']);
p.t = ["A","D","B","E","G","H","I","K","L","M","N","P","C","Q","F"];
p.s = randomArray(5,0,14);
params({co: p.co, s: p.s});
p.t0 = 'A';
p.t1 = 'B';
p.t2 = 'C';
p.t3 = 'D';
p.t4 = 'E';
p.event = function(Zone){
Zone.find('.svgedit [rx = "97"]').attr({"stroke-width": "2", "stroke": p.co[0]});
Zone.find('.svgedit path')
.attr({"stroke-width": "2", "stroke": p.co[3]})
.appendTo("svg");
};
p.mathFont = 0;Tứ giác @p.t0@@p.t1@@p.t2@@p.t3@ nội tiếp đường tròn. Biết \(\widehat{@p.t0@}[email protected]@^o;\widehat{@p.t1@}[email protected]@^o\). Tính \(\widehat{@p.t2@}-\widehat{@p.t3@}.\)
Đáp số: \(\widehat{@p.t2@}-\widehat{@p.t3@}=\) \(@p.a-p.b@^o\) || \(@p.a+p.b@^o\) || \(@p.a-p.b+10@^o\) || \(@p.a+p.b-10@^o\) .
Do tứ giác @p.t0@@p.t1@@p.t2@@p.t3@ nội tiếp nên
\(\widehat{@p.t2@}=180^o-\widehat{@p.t0@}=180^[email protected]@^[email protected]@^o\)
\(\widehat{@p.t3@}=180^o-\widehat{@p.t1@}=180^[email protected]@^[email protected]@^o\)
Vậy thì \(\widehat{@p.t2@}-\widehat{@p.t3@}[email protected]@^[email protected]@^[email protected]@^o\)
p.t = ["A","D","B","E","G","H","I","K","L","M","N","P","C","Q","F"];
p.s = randomArray(4,0,14);
p.a = random(80, 150);
p.b = random(40, p.a - 20);
params({s: p.s,a : p.a, b: p.b});
p.t0 = "A";
p.t1 = "B";
p.t2 = "C";
p.t3 = "D";
p.mathFont = 0;
Phương trình: $(x+1)^3 - x +1 = (x-1)(x-2)$ có bao nhiêu nghiệm?
$(x+1)^3 - x +1 = (x-1)(x-2)$
$⇔ x^3 + 3x^2 + 3x +1 - x + 1 = x^2 - 3x + 2$
$⇔ x^3 + 2x^2 + 5x = 0$
$⇔ x(x^2 + 2x + 5) = 0$
Phương trình có một nghiệm $x=0$.
p.a = [rand(1,-90,90,[]), rand(1,-45,45,[]), rand(1,-45,45,[]), rand(1,-45,45,[]), rand(1,-90,90,[]), rand(1,-90,90,[])];
p.g = random(9,16);
p.ko = [
'<svg width="131" height="131" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[0]+ ' 65.30556488037108,65.30558013916016) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="65.30558" cx="65.30557"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="65.06982" cx="64.8056"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m18.56374,110.24291c0,0 33.33317,-106.66614 33.33317,-106.66614c0,0 72.72691,85.80963 72.72691,85.80963"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" fill-opacity="null" fill="none" d="m45.83633,22.36455c5.45452,4.2424 18.18173,-4.2424 18.18173,-4.2424"/> <text y="87.81878" xml:space="preserve" x="57.95749" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> </g> </svg>',
'<svg width="221" height="194" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[1] + ' 110.41763305664,63.94554138183604) " id="svg_2"> <ellipse transform="rotate(5 99.32904815673783,117.21231079101555) " stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="117.21232" cx="99.32905"/> <ellipse transform="rotate(5 98.85153198242188,116.93387603759767) " stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="116.93388" cx="98.85153"/> <path transform="rotate(-117 119.8472900390625,12.205287933349604) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m66.81726,65.53836c0,0 33.33317,-106.66614 33.33317,-106.66614c0,0 72.72691,85.80963 72.72691,85.80963"/> <text y="139.84319" xml:space="preserve" x="90.72475" transform="rotate(5 100.0468749999994,131.64843749999997) " text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <path transform="rotate(5 93.93338012695321,43.43129348754887) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_4" fill-opacity="null" fill="none" d="m86.93338,37.89152c9,-5 14,12 14,12"/> </g> </svg>',
'<svg width="217" height="177.00000000000003" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[2] + '105.30556488037111,95.43053436279298) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="106.30558" cx="105.30557"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="105.06982" cx="104.8056"/> <text y="127.81878" xml:space="preserve" x="97.95749" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_4" fill-opacity="null" fill="none"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_5" fill-opacity="null" fill="none" d="m54.86385,145.7185c0,0 27.27261,-100.60562 26.95397,-100.87021c0.31864,0.26459 -37.25696,-24.58379 -37.57559,-24.84838"/> <path stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" fill-opacity="null" fill="none" d="m71.22742,37.23412c-6.06058,4.84846 7.2727,21.21203 7.2727,21.21203"/> </g> </svg>',
'<svg width="196" height="200" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[3]+ ' 108.1249923706054,98.43058013916017) " id="svg_2"> <ellipse cx="95.30557" cy="115.30558" fill="none" id="svg_1" rx="64.55558" ry="64.55558" stroke="#000" stroke-width="1.5"/> <ellipse cx="94.8056" cy="115.06982" fill="#000000" id="svg_3" rx="2" ry="2" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <text fill="#000000" fill-opacity="null" font-family="Helvetica, Arial, sans-serif" font-size="24" id="svg_7" stroke="#000" stroke-opacity="null" stroke-width="0" text-anchor="start" x="87.95749" xml:space="preserve" y="137.81878">O</text> <path d="m185.5,123.2358l-87.5,-106.2358c0.5,0.2358 55.5,124.2358 55.5,124.2358" fill="none" fill-opacity="null" id="svg_4" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <line fill="none" fill-opacity="null" id="svg_5" stroke="#000" stroke-dasharray="5,5" stroke-linecap="null" stroke-linejoin="null" stroke-opacity="null" stroke-width="1.5" x1="94.5" x2="145.5" y1="113.90246" y2="72.90246"/> <rect fill="none" fill-opacity="null" height="8" id="svg_6" stroke="#000" stroke-opacity="null" stroke-width="1.5" transform="rotate(48 144.49999999999997,79.90248107910158) " width="10" x="139.5" y="75.90248"/> <path d="m108.33337,41.49038c4.66668,4.66668 7.33335,-3.33334 7.33335,-3.33334" fill="none" fill-opacity="null" id="svg_8" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> </g> </svg>'
];
p.co = [
'<svg width="190" height="200" xmlns="http://www.w3.org/2000/svg"> <g id="svg_2" transform="rotate('+p.a[4]+' 97.57999420166037,82.55534362792973) "> <ellipse cx="81.97228" cy="89.66464" fill="none" id="svg_1" rx="64.55558" ry="64.55558" stroke="#000" stroke-width="1.5"/> <ellipse cx="81.47231" cy="89.42888" fill="#000000" id="svg_3" rx="2" ry="2" stroke="#000" stroke-opacity="null" stroke-width="1.5"/> <text fill="#000000" fill-opacity="null" font-family="Helvetica, Arial, sans-serif" font-size="24" id="svg_7" stroke="#000" stroke-opacity="null" stroke-width="0" text-anchor="start" x="47.95763" xml:space="preserve" y="102.94711">O</text> <line fill="none" fill-opacity="null" id="svg_5" stroke="#000" stroke-dasharray="5,5" stroke-linecap="null" stroke-linejoin="null" stroke-opacity="null" stroke-width="1.5" x1="81.16671" x2="132.16671" y1="88.26152" y2="47.26152"/> <rect fill="none" fill-opacity="null" height="8" id="svg_6" stroke="#000" stroke-opacity="null" stroke-width="1.5" transform="rotate(48 125.52571105957031,46.56926345825196) " width="10" x="120.52571" y="42.56926"/> <line fill="none" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" x1="101.84612" y1="10.89047" x2="177.74329" y2="103.71065" id="svg_4" stroke-linejoin="null" stroke-linecap="null" stroke="#000"/> <line fill="none" stroke="#000" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" x1="131.07679" y1="46.78777" x2="141.84598" y2="113.45421" id="svg_9" stroke-linejoin="null" stroke-linecap="null"/> <path fill="none" stroke="#000" stroke-width="1.5" stroke-opacity="null" fill-opacity="null" d="m133.64089,67.81334c4.61537,4.10255 12.82047,-3.07691 12.51245,-3.19819" id="svg_10"/> </g> </svg>',
'<svg width="185" height="173" xmlns="http://www.w3.org/2000/svg"> <g transform="rotate(' + p.a[5]+ ' 97.57999420165957,82.55534362792964) " id="svg_2"> <ellipse stroke-width="1.5" stroke="#000" ry="64.55558" rx="64.55558" id="svg_1" fill="none" cy="89.66464" cx="81.97228"/> <ellipse stroke-width="1.5" stroke-opacity="null" stroke="#000" ry="2" rx="2" id="svg_3" fill="#000000" cy="89.42888" cx="81.47231"/> <text y="102.94711" xml:space="preserve" x="47.95763" text-anchor="start" stroke-width="0" stroke-opacity="null" stroke="#000" id="svg_7" font-size="24" font-family="Helvetica, Arial, sans-serif" fill-opacity="null" fill="#000000">O</text> <line y2="47.26152" y1="88.26152" x2="132.16671" x1="81.16671" stroke-width="1.5" stroke-opacity="null" stroke-linejoin="null" stroke-linecap="null" stroke-dasharray="5,5" stroke="#000" id="svg_5" fill-opacity="null" fill="none"/> <rect y="50.34701" x="126.6368" width="10" transform="rotate(51 131.63679504394528,54.34701538085939) " stroke-width="1.5" stroke-opacity="null" stroke="#000" id="svg_6" height="8" fill-opacity="null" fill="none"/> <line stroke="#000" stroke-linecap="null" stroke-linejoin="null" id="svg_4" y2="103.71065" x2="177.74329" y1="10.89047" x1="101.84612" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> <line stroke="#000" stroke-linecap="null" stroke-linejoin="null" id="svg_9" y2="153.45408" x2="89.62391" y1="46.78777" x1="131.07683" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> <path stroke="#000" transform="rotate(-8 121.03442382812507,52.36984634399424) " id="svg_8" d="m125.40613,36.27217c-15.47044,8.66799 -5.69964,32.19535 -5.69964,32.19535" fill-opacity="null" stroke-opacity="null" stroke-width="1.5" fill="none"/> </g> </svg>'
];
params({ko: p.ko, co : p.co, a: p.a});|
Cho biết \(\widehat{\text{BAx}}\) = @2*p.g@o, \(\widehat{\text{ACB}}\) bằng:
|
function ps(numerator,denominator,check){ //hiển thị phân số, check = 0 thì hiển thị dấu trừ theo ý muốn
if(isNaN(numerator) || isNaN(denominator)) return '\\dfrac{'+numerator+'}{'+denominator+'}';
else{
var gcd = function gcd(a,b){
return Math.abs(b) ? Math.abs(gcd(b, a%b)) : Math.abs(a);
};
gcd = gcd(numerator,denominator);
var q = [numerator/gcd, denominator/gcd];
if(check != 0){
if (q[0]>0 && q[1]<0){
q[0] = q[0]*-1;
q[1] = q[1]*-1;
}else if(q[0]<0 && q[1]<0){
q[0] = q[0]*-1;
q[1] = q[1]*-1;
};
};
return (q[0] % q[1] == 0)? (q[0]/q[1]) : '\\dfrac{'+q[0]+'}{'+q[1]+'}';
};
};
p.n = randomArray(2,1,4).sort(function(a,b){return b-a});
params({n: p.n});
p.a = 2*p.n[0]+1;
p.b = 2*p.n[1]+1;Phương trình $x^2-(2m + @p.a@)x + @p.b@m = 0$ có hai nghiệm $x_1$ và $x_2$.
Giá trị nhỏ nhất của biểu thức $x_1^2+x_2^2$ là:
Phương trình có $\Delta = (2m + @p.a@)^2 - [email protected]@m = 4m^2 - @4*p.a - 4*p.b@m + @p.a*p.a@ = 4(m -@(p.a-p.b)/2@)^2 + @p.a*p.a - (p.a-p.b)*(p.a-p.b)@ >0$ nên có hai nghiệm phân biệt.
Theo hệ thức Vi-ét, ta có: $x_1 + x_2 = 2m + @p.a@$ và $x_1 . x_2 = @p.b@m$.
Do đó, $x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = 4m^2 - @4*p.a - 2*p.b@m + @p.a*p.a@ = (2m -@ps(2*p.a-p.b,2)@)^2 + @ps(4*p.a*p.a - (2*p.a-p.b)*(2*p.a-p.b),4)@ \ge @ps(4*p.a*p.a - (2*p.a-p.b)*(2*p.a-p.b),4)@$.
Dấu "=" xảy ra khi $m = @ps(2*p.a-p.b,4)@$.
require("btds");
require("mathtype");
p.toolbar = ["sqr"];
p.n = rand(2,-1,-10,[]);
p.m = rand(2,1,3,[]);
params({n: p.n, m: p.m});
p.x1 = new btds(p.n[0] + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
if(equalEq(p.x1.tex(), p.x2.tex())) p.n[0]--;
p.x1 = new btds(p.n[0] + "/" + p.m[0]);
p.x2 = new btds(p.n[1] + "/" + p.m[1]);
p.pt = new btds("(x-" + (p.x1.rutgon().tex()) + ")(x-" + (p.x2.rutgon().tex()) + ")");
Phương trình [email protected]().tex()@=0$:
Phương trình [email protected]().tex()@=0$ có $\Delta = @p.x1.cong(p.x2).rutgon().pow(2).tex()@ - [email protected](p.x2).rutgon().tex()@ > 0 $.
Mặt khác, $x_1 + x_2 = @p.x1.cong(p.x2).rutgon().tex()@ <0$ và $x_1.x_2 = @p.x1.nhan(p.x2).rutgon().tex()@ > 0$.
Suy ra phương trình có hai nghiệm cùng dấu âm.
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