### <Success>Circles: Lecture-3</Success> <div style='float: left; width:50%;'> </div>

### <Success>Recap</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>$\text { C: } x^2+y^2+2 g x+2 f y+c=0$<br> $ P:(a, b)$</p> <p>$ O P \ =\sqrt{(-g-a)^2+(-f-b)^2} \\ =\sqrt{(g^2+a^2+2 a g+f^2+b^2+2fb }$</p> <p>$O P < r \Rightarrow$ P lies inside the circle</p> <p>$O P>r \Rightarrow$ P lies outside the circle</p> <p>$O P=r \Rightarrow$ P lies on the circle</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c5981d6a-f265-4278-90f5-c6c4ef0a4e00/public"></p> <!----> </div>

### <Success>Recap</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>$OP <r$ :</p> <p>$\sqrt{g^2+a^2+2 a g+f^2+b^2+2 f b}<\sqrt{g^2+f^2-c} $</p> <p>$a^2+b^2+2 a g+2 f b+c<0$</p> <p><strong>Example:</strong></p> <p>$C: x^2+y^2+6 x-8 y+4=0 $</p> <p>P :$(2,-1)$</p> <p>O P =$\sqrt{(-5)^2+(5)^2} $</p> <p>$ =\sqrt{50}>\sqrt{21}$</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f84a602d-5bea-4c09-82d7-dd6f39403d00/public"></p> <!----> </div>

### <Success>Line and a circle</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>C: $\quad x^2+y^2+2 g x+2 f y+c=0$</p> <p>L: $\quad y=m x+d$</p> <p>let- circle $C$ and st. line L intersect at- some point $P(a, b)$.</p> <p>$ a^2+b^2+2 g a+2 f b+c=0 $</p> <p>$ b=m a+d $</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/1ac342e6-be6c-41f6-be56-2cb1cb406400/public"></p> <!----> </div>

### <Success>Line and a circle</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$ a^2+(m a+d)^2+2 g a+2 f(m a+d)+c=0 $</p> <p>$ a^2\left(1+m^2\right)+2 a(\text { md }+m f+g) +\left(a^2+2 f d+c\right)=0$</p> <p>Two distinct-real root $\left(a_0, a_1\right) ,\left(a_0, m a_0+d\right)$ $ \Rightarrow\left(a_1, m a_1+d\right)$</p> <p>Equal real roots $\left(a_0\right) \Rightarrow\left(a_0, m a_0+d\right)$</p> <p>Both roots are complex $\rightarrow$ No point of intersection.</p> </div> <div style='float: right; width:1%;'> <!----> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>C: $x^2+y^2+2 x-4 y-4=0$</p> <p>centre $(-1,2)$ and Radius = 3</p> <p>$L: x+y=-5 \Rightarrow y=-(x+5)$</p> <p>$ P(x, y) $</p> <p>$x^2+(x+5)^2 +2 x+4(x+5)-4=0$</p> <p>$2 x^2+16 x+41=0 $</p> <p>$x=\frac{-16 \pm \sqrt{(256-328)}}{4}$</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f960675d-2e16-48b0-fc03-0e7ba19e3700/public"></p> <!----> </div>

#### <Success>Intercepts made by a circle on the axes </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> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/7444d059-c9b1-48fd-86dc-c2ba3349c700/public"> <!----> </div>

#### <Success>Intercepts made by a circle on the axes </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>$a^2+2 g a+c=0$</p> <p>$a_0 , a_1=-g \pm \sqrt{g^2-c}$</p> <p>$g^2<c$ then the circle does not intersect - $x$-axis. (No intercept $x$-axis)</p> <ul> <li>if $g^2=c$ then the circle “touches” the $x$-axis at the point $(-g,0)$.</li> </ul> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/3e010f93-67d5-4583-7753-512f82066000/public"> <!----> </div>

#### <Success>Intercepts made by a circle on the axes </Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>if $g^2>c$,</p> <p>$a_0, a_1$</p> <p>$ a_0=-g+\sqrt{g^2-c} $</p> <p>$ a_1=-g-\sqrt{g^2-c}$</p> <p>the circle intersect the $x$-axis at two distinct points</p> <p>$\left(-g+\sqrt{g^2-c}, 0\right),\left(-g-\sqrt{g^2-c}, 0\right)$</p> <p>$\therefore$ intercept made by the circle on the $x$-axis $=2 \sqrt{g^2-c}$.</p> </div> <div style='float: right; width:1%;'> <!----> </div>

#### <Success>Intercepts made by a circle on the axes </Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <ul> <li> <p>if $f^2>c$ then the intersect made by the circle on the y-axis is $=2 \sqrt{f^2-c}.$</p> </li> <li> <p>if $f^2=c$ then the intersect made by the circle on the y -axis is $0.$</p> </li> <li> <p>if $f^2<c$ then the circle does not intersect with the y-axis.</p> </li> </ul> </div> <div style='float: right; width:1%;'> <!----> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>If one of the diameters of the circle $x^2+y^2-2 x-6 y+6=0$ is a chord to some other circle centered at $(2,1)$ then the radius of the other circle is <em><strong><span style="color:blue">3</span></strong></em>_.</p> </div> <div style='float: right; width:30%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/05f63e06-7545-4891-f691-d8697cd57c00/public"> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/6b600b4c-cfdb-4a4a-3771-dc4cabbbe500/public"> <!----> </div>

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