### <Success>Circles: Lecture 4</Success> <div style='float: left; width:50%;'> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>Find the equation of circle (s) touching the $x$-axis at a distance of 3 units from the origin and having an intercept of length $2\sqrt{7}$ on the Y-axis?</p> </div> <div style='float: right; width:1%;'> <!----> </div>

### <Success>Solution</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>Let $\quad x^2+y^2+2 y x+2 y+c=0$ the only point-which lies both the circle and the $x$-axis is $(3,0)$</p> <p>$(a,0)$</p> <p>$ a^2+2 g a+c=0 $</p> <p>$ \Rightarrow \quad g^2=c $</p> <p>$ 9+6 g+c=0 $</p> <p>$ 9+6 g+g^2=0 \Rightarrow(g+3)^2=0$<br> $\Rightarrow c = 9 ,g=-3 $</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9e629349-c078-4dd0-b5e2-ff1b45f8bf00/public"> <!----> <!----> </div>

### <Success>Solution</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$\begin{gathered}2 \sqrt{f^2-c}=2 \sqrt{f^2-9}=2\sqrt{7} \\ f= \pm 4 .\end{gathered}$</p> <p>$C_1: \ $ $ \ g=-3, f=-4$</p> <p>Centre $(-g , -f) = (3 , 4)$</p> <p>$ C_2: g=-3, f=+4$</p> <p>Centre $(-g , -f) = (3 , -4 )$</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ea850a94-65e1-4904-8eaa-3f706624f100/public"> <!----> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>Let $L_1$, be a straight line passing through the origin and $L_2$ be the straight line $x+y=1$. If the intercepts made by the circle $x^2+y^2-x+3 y=0$ on $L_1$ and $L_2$ are equal, then which of the following equation can represent $L_1$ ?</p> <p>a) $x+y=0$</p> <p>b) $x-y=0$</p> <p>c) $x+7 y=0$</p> <p>d) $x-7 y=0$</p> </div> <div style='float: right; width:30%;'> <!----> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f1e7591d-8971-48ad-7a6d-c5ebb31e6f00/public"> <!----> </div>

### <Success>Solution</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$\begin{aligned} C: \quad x^2+y^2-x & +3 y=0 \\ L_2: \quad x+y & =1 \\ y & =1-x\end{aligned}$</p> <p>$x^2+(1-x)^2-x+3(1-x)=0 $</p> <p>$ \Rightarrow \quad x^2-3 x+2=0 $</p> <p>$\Rightarrow \quad x=1,2$</p> <p>$x^2+m^2 x^2-x+3 m x=0 $</p> <p>$x^2\left(1+m^2\right)-(1-3 m) x=0 $</p> <p>$x=0,(1-3 m) /\left(1+m^2\right)$</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/38312de4-8471-4a0c-9053-8bc85687aa00/public"> <!----> </div>

### <Success>Solution</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$\therefore L_1$ intersects the circle at- $(0,0)$ $(y=mx)$ and $\left(\frac{1-3 m}{1+m^2}, \frac{m(1-3 m)}{1+m^2}\right)$</p> <p>$\therefore \sqrt{2}=\sqrt{\frac{(1-3 m)^2}{\left(1+m^2\right)}}$</p> <p>$\Rightarrow \quad 2=(-3 m)^2 /\left(1+m^2\right) $</p> <p>$\Rightarrow \quad 2+2 m^2=1-6 m+9 m^2 $</p> <p>$\Rightarrow \quad 7 m^2-6 m-1=0 $</p> <p>$\Rightarrow \quad(m-1)(7 m+1)=0$</p> <p>$\Rightarrow m=1,-\frac{1}{7}$.</p> <p>$\therefore L_1, y=x $ or $\ y=\frac{-x}{7}$.</p> </div> <div style='float: right; width:1%;'> <!----> <!----> </div>

### <Success>Equation of the tangent to a circle at a given point</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$ C: x^2+y^2+2 g x+2 f y+c=0$<br> $ P\left(x_1, y_1\right)$</p> <p>Slope of tangent = $ \frac{(y-y_1)}{(x-x_1)}$</p> <p>$\frac{y_1-(-f)}{x_1-(-g)}=\frac{y_1+f}{x_1+g}=\text { slope of OP} $</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f313a0d6-6ed6-477c-c3e1-1e185ac30c00/public"> <!----> </div>

### <Success>Equation of the tangent to a circle at a given point</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$ \frac{\left(y-y_1\right)}{\left(x-x_1\right)} \times \frac{\left(y_1+f\right)}{\left(x_1+y\right)}=-1$</p> <p>$ \Rightarrow \ \left(y-y_1\right)\left(y_1+f\right)+\left(x-x_1\right)\left(x_1+g\right)=0 $</p> <p>$ \Rightarrow y y_1 +x x_1 +y f+x g= y_1^2 + y_1 f+ x_1^2 + x_1 g $</p> <p>since $P\left(x_1, y_1\right)$ lies on circle $C$,</p> <p>$ x_1^2+y_1^2+2 g x_1+2 f y_1+c=0 \\ x_1^2+y_1^2+g x_1+f y_1={-\left(g x_1+f y_1\right.}+c)$</p> </div> <div style='float: right; width:1%;'> <!----> </div>

### <Success>Equation of the tangent </Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$ y y_1+x x_1+y f+x g+g x_1+f y_1+c=0$</p> <p>$y\left(y_1+f\right)+x\left(x_1+g\right)+g x_1+f y_1+c=0$</p> </div> <div style='float: right; width:1%;'> <!----> </div>

#### <Success>Equation of the normal to a circle at a point on the circle</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$ C: x^2+y^2+2 g x+2 fy+c=0 $</p> <p>$\text { slope } of \text { the normal at } P $</p> <p>$ =\left(y_2-(-f)\right) /\left(x_2-(-g)\right) $</p> <p>$=(y-(-f)) /(x-(-g))$</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f1123b48-1c6c-4859-2fbf-738ff8551b00/public"> <!----> </div>

#### <Success>Equation of the normal to a circle at a point on the circle</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$\frac{(\left.y_2+f\right)}{\left(x_2+g\right)}=\frac{(y+f)}{(x+g)}$</p> <p>$\Rightarrow \ x\left(y_2+f\right)+g\left(y_2+f\right)=y\left(x_2+g\right)+f\left(x_2+g\right)$</p> <p>$\text { i.e., } \ x\left(y_2+f\right)-y\left(x_2+g\right)$</p> <p>$ =f\left(x_2+g\right) -g(y_2+f) $</p> <p>$ =\left(f x_2-g y_2\right)$</p> </div> <div style='float: right; width:1%;'> <!----> </div>

#### <Success>Length of the tangent to a given circle from a given point</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$C: x^2+y^2+2 g x+2 f y+c=0 $</p> <p>$ O P= \sqrt{\left(x_1-(-g)\right)^2+\left(y_1-(-f)\right)^2} $</p> <p>$ O P=\sqrt{\left(x_1+g\right)^2+\left(y_1+f\right)^2}$</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/37b8d82f-d4f7-45a9-0ab8-e071df6c7500/public"> <!----> </div>

#### <Success>Length of the tangent to a given circle from a given point</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$ \because O P^2=O T^2+P T^2 $</p> <p>$\text { i.e., } $</p> <p>$P T=\sqrt{O P^2-O T^2} $</p> <p>$ P T=\sqrt{\left(x_1+g\right)^2+\left(y_1+f\right)^2-\left(g^2+f^2-c\right)}$</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/04fe3378-797d-4a76-4838-08240bc50a00/public"> <!----> <!----> </div>

<div> <Success><h2>Thank you</h2></Success> </div>