mathematics-class-11-unit-12-chapter-21-conic-section-l-8-10-4r4r1tkhrbc

<h4 id="successpoints-of-intersection-of-line-with-ellipsesuccess"><Success>Points of intersection of line with ellipse</Success></h4>
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<p>Points of intersection of line $y=m x+c$ with the ellipse</p>
<p>$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</p>
<p>Putting $y=m x+c$ in the equation of ellipse gives</p>
<p>$ \frac{x^2}{a^2}+\frac{(m x+c)^2}{b^2}=1 $</p>
<p>$\Leftrightarrow  b^2 x^2+a^2\left[m^2 x^2+2 m c x+c^2\right]=a^2 b^2 $</p>
<p>$\left(a^2 m^2+b^2\right) x^2+2 a^2 m c x+a^2\left(c^2-b^2\right)=0$</p>
<p>This is a quadratic equation in $x$ with discriminant $D=\left(2 a^2 m c\right)^2-4\left(a^2 m^2+b^2\right) a^2\left(c^2-b^2)\right.$</p>
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<h4 id="successpoints-of-intersection-of-line-with-ellipsesuccess-1"><Success>Points of intersection of line with ellipse</Success></h4>
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<p>that is $ D=4 a^4 m^2 c^2-4 a^4 a^2 c^2+ $</p>
<p>$ 4 a^4 m^2 b^2-4 a^2 b^2 c^2  +4 a^2 b^4 $</p>
<p>$=4 a^2 b^2\left(a^2 m^2-c^2+b^2\right)$</p>
<p>Two points of intersection if $D>0$ that is</p>
<p>$c^2<a^2 m^2+b^2$</p>
<p>======</p>
<h4 id="successpoints-of-intersection-of-line-with-ellipsesuccess-2"><Success>Points of intersection of line with ellipse</Success></h4>
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<p>One points of intersection if $D=0$ that is</p>
<p>$c^2=a^2 m^2+b^2$</p>
<p>No points of intersection if $D<0$ that is $c^2>a^2 m^2+b^2$</p>
<p>conclusion: The line $y=m x+c$ is tangent to the ellipse</p>
<p>$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</p>
<p>if and only if $c^2=a^2 m^2+b^2$</p>
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<h3 id="successlength-of-the-chordsuccess"><Success>Length of the chord</Success></h3>
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<p>Length of the chord joining $\left(x_1, y_1\right)$ and $\left(x_2, y\right.$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</p>
<p>The  equation of the line joining $(x_1, y_1)$ & $(x_2, y_2)$ is</p>
<p>$
y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)
$</p>
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<h3 id="successlength-of-the-chordsuccess-1"><Success>Length of the chord</Success></h3>
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<p>that is $y=\underbrace{\left(\frac{y_2-y_1}{x_2-x_1}\right)}_m x+\underbrace{y_1-\left(\frac{y_2-y_1}{x_2-x_1}\right)}_c x_1$</p>
<p>$x_1$ and $x_2$ are the roots of the quadratic equation.</p>
<p>$ \quad\left(a^2 m^2+b^2\right) x^2+2 a^2 m c x+a^2\left(c^2-b^2\right)=0 $</p>
<p>$ \therefore \quad x_1+x_2=-\frac{2 a^2 m c}{a^2 m^2+b^2} ; x_1 x_2=\frac{a^2\left(c^2-b^2\right)}{a^2 m^2+b^2}$</p>
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<h4 id="successequation-of-the-tangent-to-the-ellipsesuccess-1"><Success>Equation of the tangent to the ellipse</Success></h4>
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<p>that is $ D  =4\left(a^2 y_1^2+b^2 x_1^2-a^2 b^2\right)$</p>
<p>$ =4 a^2 b^2\left(\frac{x_1^2}{a^2}+\frac{y_1^2}{b}-1\right)=0 $</p>
<p>$\therefore \quad m  =\frac{2 x_1 y_1}{2\left(x_1^2-a^2\right)}=\frac{x_1 y_1}{x_1^2-a^2} .$</p>
<p>$(\therefore (x_1, y_1)$ lies on the ellipse)</p>
<p>So, the equation of the tangent line is</p>
<p>$\Leftrightarrow y-y_1=\frac{x_1 y_1}{x_1^2-a^2}(x-x_1)$</p>
<p>$\Leftrightarrow y-y_1=\frac{x_1 y_1}{x_1^2-a^2} x-\frac{x_1^2 y_1}{x_1^2-a^2}$</p>
<p>$ \Leftrightarrow y y_1-y_1^2=\frac{x_1 y_1^2}{x_1^2-a^2} x-\frac{x_1^2 y_1^2}{x_1^2-a^2} $</p>
<p>$ \Leftrightarrow y y_1=\frac{y_1^2}{x_1^2-a^2}\left(x x_1\right)+y_1^2 \left[1-\frac{x_1^2}{x_1^2-a^2}\right]$</p>
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<h4 id="successequation-of-the-tangent-to-the-ellipsesuccess-3"><Success>Equation of the tangent to the ellipse</Success></h4>
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<p><strong>Another way using Calculas</strong></p>
<p>$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$</p>
<p>Differentating w.r.t. $x_1$ we get</p>
<p>$\frac{2 x}{a^2}+\frac{2 y \frac{d y}{d x}}{b^2}=0 \Rightarrow \frac{d y}{d x}=-\frac{b^2}{a^2} \frac{x}{y}$</p>
<p>Recall : slope of tangent to a curve $y=f(x)$ at point $(x_1 , y_1)$ is $m=\left.\frac{d y}{d x}\right|_{(x, y)}$</p>
<p>$\therefore m=\frac{-b^2}{a^2} \frac{x_1}{y_1}$.</p>
<p>$\therefore$ Equation of the tangent is $y-y_1=\frac{-b^2}{a^2} \frac{x_1}{y_1}\left(x-x_1\right)$</p>
<p>that is $y-y_1=-\frac{b^2}{a^2} \frac{x_1}{y_1} x+\frac{b^2}{a^2} \frac{x_1^2}{y_1}$</p>
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