mathematics-class-11-unit-12-chapter-12-circles-l-12-13-2qvguvuh6mo

### <Success>Solution</Success>
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<p>$S^{\prime}=0$</p>
<p>$x^2+y^2+2g^{\prime}x+2f^{\prime}+c^{\prime}=0$</p>
<p>$L=mx+ny+p=0$</p>
<p>Radical axis between $S=0$ and $s^{\prime}=0$ muSt be $L=0,$</p>
<p>$ S=x^2+y^2+2 g x+2 f y+c=0$</p>
<p>$S-S^{\prime}=0$, ie;</p>
<p>$S-S^{\prime}=2\left(g-g^{\prime}\right) x+2\left(f-f^{\prime}\right) y+c-c^{\prime}=0 $</p>
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<p>======</p>
<h3 id="successsolutionsuccess"><Success>Solution</Success></h3>
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<p>$L=m x+n y+p=0 \quad \quad  \times q$</p>
<p>$m q x + nqy +pq=0$</p>
<p>$m q=2\left(g-g^{\prime})\Rightarrow 2 g=2 g^{\prime}+m q\right. $</p>
<p>$n q=2\left(f-f^{\prime}\right) and / \Rightarrow 2 f=2 f^{\prime}+n q $</p>
<p>$p q=c-c^{\prime} \Rightarrow c=c^{\prime}+p q $</p>
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<p>======</p>
<h3 id="successsolutionsuccess-1"><Success>Solution</Success></h3>
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<p>$S=x^2+y^2+2 g x+2 f y+c=0 $</p>
<p>$S=x^2+y^2+\left(2 g^{\prime}+m q\right) x+\left(2 f^{\prime}+n q\right) y+ c^{\prime} + pq=0$</p>
<p>$\left(x^2+y^2+2 g^{\prime} x+2 f^{\prime} y+c^{\prime}\right)+q(m x+n y +p)=0$</p>
<p>$S=\left( S^{\prime}+q L=0\right)$</p>
<p>$ \left[{\because S^{\prime} = \left(x^2+y^2+2 g^{\prime} x+2 f^{\prime} y+c^{\prime}\right), \& \quad L = (m x+n y +p)}\right]$</p>
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##### <Success>General equation for family of circles which pass through the any two points</Success>
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<p>$S^{\prime}:\left(x-\frac{x_1+x_2}{2}\right)^2 +\left(y-\frac{y_1-y_2}{2}\right)^2 -\frac{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}{4}=0$</p>
<p>$\frac{y-y_1}{x-x_1}=\frac{y_1-y_2}{x_1-x_2}$</p>
<p>$L\left(x-x_1\right)\left(y_1-y_2\right)+\left(y-y_1\right)\left(x_2-x_1\right)=0 $</p>
<p>$S=S^{\prime}+K L=0 \quad(K \text { is real })$</p>
<p>$S=\left(x-\frac{x_1+x_2}{2}\right)^2+\left(y-\frac{y_1+y_2}{2}\right)^2-\left[\frac{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}{4}\right] +k\left(\left(x-x_1\right)\left(y_1-y_2\right)+\left(y-y_1\right)\left(x_2-x_1\right)\right)=0 $</p>
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