### <Success>Circles: Lecture-6</Success> <div style='float: left; width:50%;'> </div>

### <Success>Director circle</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <ul> <li>Centre of the director circle to a given circle is same as the centre of the given circle.</li> <li>Radius of the director circle is $\sqrt{2}$ times the radius of the given circle.</li> </ul> </div> <div style='float: right; width:50%;'> <img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f18bbdaa-57b0-408b-dd7e-85bf5ebf8a00/public"> <!----> </div>

### <Success>Shortest- distance of a point-from a straight line</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>$ d^2=\frac{\left[m\left(x_0-\alpha\right)-\left(y_0-\beta\right)\right]^2}{\left(1+m^2\right)}$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/6a12c133-6e68-4fbd-9132-ebf9675ed800/public"></p> <!----> </div>

#### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p><ins><strong>CASE-I</strong></ins></p> <p>Circles neither touch nor intersect</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/43789de5-e89a-419e-d5b5-6561a7ca3400/public"></p> <!----> </div>

#### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>$ C_1:\left(x_1, y_1\right)$</p> <p>$ C_2: \left(x_2, y_2\right)$</p> <p>$ \sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2} >r_1+r_2 $</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/618b59b3-f9d2-4a31-b49f-3d54f9658300/public"></p> <!----> </div> <p>======</p> <h4 id="successcommon-tangents-between-two-circlessuccess"><Success>Common tangents between two circles</Success></h4> <div style='float: left; text-align:left; width:50%;font-size:25px;'> <p>$\triangle P B C_2 \sim \triangle P A C_1$</p> <p>$ \frac{P c_1}{P c_2}=\frac{r_1}{r_2}$</p> <p>$ \frac{l_1+l_2}{l_2}=1+\frac{l_1}{l_2}=\frac{r_1}{r_2}$</p> <p>$ l_1 / l_2=\left(r_1-r_2\right) / r_2$</p> <p>$ \Rightarrow \quad l_2=\frac{l_1 r_2}{\left(r_1-r_2\right)} \Rightarrow l_2^2=\frac{l_1^2 r_2^2}{\left(r_1-r_2\right)^2}$</p> <p>$ \frac{\left(\beta-y_1\right)}{\left(\alpha-x_1\right)}=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$</p> <p>$ \quad\quad\quad\quad =\frac{\left(\beta-y_2\right)}{\left(\alpha-x_2\right)}$</p> <p>$l_2^2=\left(\beta-y_2\right)^2 +\left(\alpha-x_2\right)^2$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/51f691d6-7871-4878-fead-b2ea67557100/public"></p> </div>

#### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$ l_2^2=\left(\beta-y_2\right)^2+\left(\alpha-x_2\right)^2=\left(\alpha-x_2\right)^2 \times{\left[1+\frac{\left(\beta-y_2\right)^2}{\left(\alpha-x_2\right)^2}\right]}$</p> <p>$\Rightarrow l_2^2=\left(\alpha-x_2\right)^2\left[1+\frac{\left(y_2-y_1\right)^2}{\left(x_2-x_1\right)^2}\right]$</p> <p>$=\frac{l_1^2 r_2^2}{\left(r_1-r_2\right)^2}=\frac{r_2^2}{\left(r_1-r_2\right)^2}\left[\left(y_2-y_1\right)^2+\left(x_2-x_1\right)^2\right]$</p> <p>$\frac{(\alpha-x_2 )^2}{ (x_2-x_1 )^2} [ (x_2-x_1 )^2+ (y_2-y_1)^2 ]$$ $$=\frac{r_2^2}{ (r_1-r_2 )^2} \times [ (x_2-x_1 )^2+ (y_2-y_1)^2 ]$</p> <p>$\alpha-x_2=\frac{r_2}{\left(r_1-r_2\right)}\left(x_2-x_1\right) $</p> </div> <div style='float: right; width:1%;'> <!----> </div>

#### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>$\alpha= x_2+\frac{r_2\left(x_2-x_1\right)}{\left(r_1-r_2\right)}$</p> <p>$\alpha= \left(r_1 x_2-r_2 x_1\right) /\left(r_1-r_2\right)$</p> <p>$\beta= \left(r_1 y_2-r_2 y_1\right) /\left(r_1-r_2\right)$</p> <p>$\frac{(y-\beta)}{(x-\alpha)} = m $</p> <p>$\frac{\left(\beta -y_1\right)}{\left(\alpha-x_1\right)}=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/434cd04c-3346-4db3-55d5-947fc9978400/public"></p> <!----> </div>

<h3 id="successcommon-tangents-between-two-circlessuccess-1"><Success>Common tangents between two circles</Success></h3> <div style='float: left; text-align:left; width:80%;font-size:30px;'> <p>As Shortest- distance of a point-from a straight line:</p> <p>$d^2=\frac{\left[m\left(x_0-\alpha\right)-\left(y_0-\beta\right)\right]^2}{\left(1+m^2\right)}$</p> <p>$\Rightarrow \frac{\left[m\left(x_1-\alpha\right)-\left(y_1-\beta\right)\right]^2}{\left(1+m^2\right)}=r_1^2 \ldots(i)$</p> <p>$\frac{\left[m\left(x_2-\alpha\right)-\left(y_2-\beta\right)\right]^2}{\left(1+m^2\right)}=r_2^2 \ldots(ii)$</p> <p>$ (x_2-\alpha)^2 \frac{\left[m-\frac{(y-\beta)}{(x_2-\alpha)}\right]^2}{(1+m^2)} =r_2^2 $</p> <p>$\frac{\left[m-\frac{(y_2-\beta)}{(x_2-\alpha)}\right]^2}{[1+m^2]}=\frac{r_2^2}{(x_2-\alpha)^2}\ldots({ii}^{\prime})$</p> </div> <div style='float: right; width:1%;'> <!----> <!----> <!----> </div>

### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$ \frac{\left[m-\frac{ (y_1-\beta )}{ (x_1-\alpha )} \right]^2}{ (1+m^2 )}=\frac{r_1^2}{ (x_1-\alpha )^2} \ldots (i^{\prime})$</p> <p>$\frac{ \left[m-\frac{(y-\beta)}{ (x_1-\alpha )} \right]^2}{ (1+m^2 )}=\frac{r_2^2}{ (x_2-\alpha)^2} \ldots (ii^{\prime})$</p> <p>$\frac{r_1}{r_2}=\frac{P C_1}{P C_2} \Rightarrow \frac{r_1^2}{r_2^2}=\frac{P C_1^2}{P C_2^2}$</p> <p>$\frac{r_1^2}{r_2^2}=\frac{\left(x_1-\alpha\right)^2+\left(y_1-\beta\right)^2}{\left(x_2-\alpha\right)^2+\left(y_2-\beta\right)^2}=\frac{\left(x_1-\alpha\right)^2\left[1+\frac{\left(y_1-\beta\right)^2}{\left(x_1-\alpha\right)^2}\right]}{\left(x_2-\alpha\right)^2\left[1+\frac{\left(y_2-\beta\right)^2}{\left(x_2-\alpha\right)^2}\right]}$</p> </div> <div style='float: right; width:1%;'> <!----> </div>

### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$s$ : slope of line joining $C_1$ and $C_2$.</p> <p>$\frac{(m-s)^2}{\left(1+m^2\right)}=\frac{r_1^2}{\left(x_1-\alpha\right)^2}, \alpha=\frac{r_1 x_2-r_2 x_1}{\left(r_1-r_2\right)}$</p> <p>$\therefore x_1-\alpha=x_1+\frac{\left(r_2 x_1-r_1 x_2\right)}{\left(r_1-r_2\right)}$</p> <p>$\text { ie., } x_1-\alpha=\frac{r_1\left(x_1-x_2\right)}{\left(r_1-r_2\right)}$</p> <p>$ \frac{(m-s)^2}{\left(1+m^2\right)}=\frac{\left(r_1-r_2\right)^2}{\left(x_1-x_2\right)^2}=k$</p> <p>$ m=m_1,m_2$</p> <p>$ m^2-2 m s+s^2=k+k m^2$</p> </div> <div style='float: right; width:1%;'> <!----> </div>

### <Success>Common tangents between two circles</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>$ m^2(k-1)+2 ms+\left(k-s^2\right)=0$</p> <p>$m_1=m_1, m_2 \cdot$</p> <p>$(y-\beta)=m_1(x-\alpha),$<br> $\quad(y-\beta)=m_2(x-\alpha)$</p> <p>$\frac{P C_1}{P C_2}=\frac{r_1}{r_2}$</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/48fe18e3-c8f2-4398-ea2d-72d455ce4300/public"></p> <!----> <!----> <!----> <!----> </div>

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