mathematics-class-11-unit-12-chapter-13-circles-l-13-13-zhue3nx1vdy

##### <Success>General equation for family of circles: straight line is parallel to the y-axis</Success>
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<p>$(x+g)^2+(y-y_1)^2=\left(x_1+g\right)^2 $</p>
<p>$x^2+2 g x+g^2 +\left(y-y_1\right)^2 =x_1^2+g^2+2 g_{x_1}$</p>
<p>$x^2+\left(y-y_1\right)^2+2 g\left(x-x_1\right)-x_1^2=0 $</p>
<p>$\left(x-x_1\right)^2+2 x x_1-x_1^2+\left(y-y_1\right)^2+2 g\left(x-x_1\right)-x_1^2=0$</p>
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<p>======</p>
<h5 id="successgeneral-equation-for-family-of-circles-straight-line-is-parallel-to-the-y-axissuccess"><Success>General equation for family of circles: straight line is parallel to the y-axis</Success></h5>
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<p>$\left(x-x_1\right)^2+\left(y-y_1\right)^2+2 g\left(x-x_1\right)+ 2xx_1 - 2x_1^2 = 0 $</p>
<p>$\left(x-x_1\right)^2+\left(y-y_1\right)^2+\left(x-x_1\right)({2 g}+2 x_1)=0 $</p>
<p>$\left(x-x_1\right)^2+\left(y-y_1\right)^2+k\left(x-x_1\right)=0$</p>
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##### <Success>General equation for family of circles: straight line is parallel to the x-axis</Success>
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<p>$\left(x-x_1\right)^2+(y+f)^2=\left(y_1+f\right)^2 $</p>
<p>$\left(x-x_1\right)^2+\left(y-y_1\right)^2+k\left(y-y_1\right)=0 $</p>
<p>$\quad$</p>
<p>$ x^2+y^2+2 x\left(-x_1-m y_1\right) +f(2 y-2 m x) -x_1^2-y_1^2+ 2 x_1\left(x_1+m\left(y_1+f\right)\right)-2 f y_1=0  $</p>
<p>$\quad$</p>
<p>$(x-x_1)^2+(y-y_1)^2+K[(y-y_1)-m(x-x_1)] = 0$</p>
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<h5 id="successgeneral-equation-for-family-of-circles-straight-line-is-parallel-to-the-x-axissuccess"><Success>General equation for family of circles: straight line is parallel to the x-axis</Success></h5>
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<p>$\left(x-x_1\right)^2+2 x x_1-x_1^2+\left(y-y_1\right)^2+2 y y_1-y_1^2 -2x x_1-2 m xy_1+2 fy-2 m fx -x_1^2-y_1^2+2 x_1^2+2 m x_1\left(y_1+f\right)-2 f y_1=0 $</p>
<p>$\quad$</p>
<p>$\left(x-x_1\right)^2+\left(y-y_1\right)^2+2 y_1\left(y-y_1\right) -2 y_1 m\left(x-x_1\right)+2 f\left(y-y_1\right) -2 m f\left(x-x_1\right)=0$</p>
<p>$\quad$</p>
<p>$ (x-x_1)^2+ (y-y_1)^2 +2 y_1 ((y-y_1)-m (x-x_1)) +2 f ((y-y_1)-m(x-x_1)) =0 $</p>
<p>$(x-x_1)^2+(y-y_1)^2+2(y_1+f)((y-y_1)-m(x-x_1))=0$</p>
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### <Success>Equation for the chord of contact</Success>
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<p>$x^2+y^2+2 g x+2 f y+c=0 $</p>
<p>$R=\sqrt{g^2+f^2-c}  $</p>
<p>$P O^2=L^2+R^2 $</p>
<p>$P O^2=\left(x_1+g\right)^2+(y_1+f)^2 $</p>
<p>$L=\sqrt{P O^2-R^2} $</p>
<p>$=\sqrt{\left(x_1+g\right)^2+\left(y_1+f\right)^2-R^2}$</p>
<p>$ L^2 = x_1^2+ 2 g x_1 + y_1^2 +2 f y_1 + g^2 +f^2 - R^2 $</p>
<p>$S_1=\left(x-x_1\right)^2+\left(y-y_1\right)^2-L^2=0 $</p>
<p>$S_2=x^2+y^2+2 g x+2 f y+c=0 $</p>
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<h3 id="successequation-for-the-chord-of-contactsuccess"><Success>Equation for the chord of contact</Success></h3>
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<p>$S_1-S_2=0 $</p>
<p>$x^2+y^2+2 g x+2 f y+c-\left(x-x_1\right)^2 $</p>
<p>$-\left(y-y_1\right)^2+L^2 =0$</p>
<p>$2 x x_1-x_1^2+2 yy_1-y_1^2+2 g x+2 f y+c+L^2=0 $</p>
<p>i.e:</p>
<p>$2 x\left(g+x_1\right) +2 y\left(f+y_1\right) -\left(x_1^2+y_1^2-L^2-c\right)=0 .$</p>
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<h3 id="successlength-of-chordsuccess"><Success>Length of chord</Success></h3>
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<p>OM = h
$\triangle T_1 M O \sim \triangle T_1 P O$</p>
<p>$\frac{x}{L}=\frac{h}{R}=\frac{R}{\sqrt{R^2+L^2}}$</p>
<p>$x=\frac{R L}{\sqrt{R^2+L^2}},$
$h=\frac{R^2}{\sqrt{R^2+L^2}}$<br>
$T_1 T_2=\frac{2 R L}{\sqrt{R^2+L^2}}$</p>
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<h3 id="successarea-of-triangle-success"><Success>Area of triangle </Success></h3>
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<p>$\operatorname{Area}\left(P T_1 O T_2\right)=L R $</p>
<p>$T_2$ Area $\left(O T_1 T_2\right)=h x$</p>
<p>$=\frac{R^3 L}{\left(R^2+L^2\right)}$</p>
<p>$\therefore$ Area $\left(\triangle P T_1 T_2\right)$</p>
<p>$=\text { Area }\left(P T_1 O T_2\right)-\text { Area } (OT_1T_2) $</p>
<p>$=L R-\left(R^3 L\right) /\left(R^2+L^2\right) =R L^3 /\left(R^2+L^2\right)$</p>
<p>$\angle T_1 P T_2 =2 \tan ^{-1}(R / L) \\ =\tan ^{-1}(R / L)+\tan ^{-1}(P/ L) \\ =\tan ^{-1}\left[\frac{(2 R / L)}{1-R^2 / L^2}\right]$</p>
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