### <Success>Circles: lecture-11</Success> <div style='float: left; width:50%;'> </div>

##### <Success>Two circles intersect each other at two points</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> $S_1 = 0$ and $S_2 = 0$ <p>$(r_1,o_1)$ and $(r_2,o_2)$</p> <p>$|r_1 - r_2|<d_{o_1,o_2}<r_1+r_2$ (two circles intersect at 2 points)</p> <p>$d_{o_1,o_2}=r_1+r_2$ (two circles touch each other externally)</p> <p>$d_{o_1,o_2} = |r_1 - r_2|$ (two circles touch each other internally)</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-12-chapter-11-circles-l-11_13-fdjwbn2dkpm/img/031-0241.2.jpg"></p> </div>

##### <Success>Two circles intersect each other at two points</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> $r_1 \ge r_2$ <p>$x=a+r_1 \cos \theta$</p> <p>$y=b+r_1 \sin \theta$</p> <p>$x=c+r_2 \cos \phi$</p> <p>$y=d+r_2 \sin \phi$</p> <p>$a+r_1 \cos \theta = c+r_2 \cos \phi$</p> <p>$b + r_1 \sin \theta=d+r_2 \sin \phi$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/bed5032a-0735-476b-4744-064f54bf9400/public"></p> </div> <p>======</p> <h5 id="successtwo-circles-intersect-each-other-at-two-pointssuccess"><Success>Two circles intersect each other at two points</Success></h5> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$r_1 \cos \theta = (c-a)+r_2 \cos \phi,$<br> $r_1 \sin \theta = (d-b) + r_2 \sin \phi$</p> <p>$r_1^2 \cos^2 \theta + r_1^2 \sin^2 \theta = (c-a)^2 + r_2^2 \cos^2 \phi + 2 (c-a)r_2 \cos \phi + (d-b)^2 + r_2^2 \sin^2 \phi + 2(d-b) r_2 \sin \phi$</p> </div>

##### <Success>Two circles intersect each other at two points</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$r_1^2 = (c-a)^2 + (d-b)^2 + r_2^2 +2r_2 d_{o_1,o_2} \biggl(\frac{(c-a)}{d_{o_1,o_2}} \cos \phi + \frac{(d-b)}{d_{o_1,o_2}}\sin\phi \biggl)$</p> <p>$r_1^2 = r_2^2+d_{o_1,o_2}^2 + 2r^2 d_{o_1,o_2} (\cos \alpha \cos \phi + \sin \alpha \sin \phi)$</p> <p>$r_1^2 = r_2^2 + d_{o_1,o_2}^2 + 2r_2 d_{o_1,o_2} \cos (\phi - \alpha)$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-12-chapter-11-circles-l-11_13-fdjwbn2dkpm/img/118-0946.8.jpg"></p> </div>

##### <Success>Two circles intersect each other at two points</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>$\cos (\phi - \alpha)=\frac{r_1^2 - (r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2f05e195-211e-4902-eb03-786f8a1d3900/public"></p> </div>

##### <Success>Two circles touch each other externally</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> $\frac{r_1^2 - (r_2^3 + d_{o_1,o_2})^2}{2r_2 d_{o_1,o_2}}=-1 = \cos(\phi - \alpha)$ <p>$\Rightarrow \phi - \alpha = \pi$</p> <p>$\Rightarrow \phi = \alpha + \pi$</p> <p>$r_1^2 = r_2^2 + d_{o_1,o_2}^2 - 2r_2 d_{o_1,o_2}$</p> <p>$\Rightarrow r_1^2 = (r_1 - d_{o_1,o_2})^2$</p> <p>$\Rightarrow r_1 = r_2 - d_{o_1,o_2}$, or $r_1 = d_{o_1,o_2}-r_2$</p> <p>$d_{o_1,o_2}=r_1+r_2$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-12-chapter-11-circles-l-11_13-fdjwbn2dkpm/img/203-1663.2.jpg"></p> <!----> </div>

##### <Success>Two circles touch each other externally</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> $d_{o_1,o_2}=r_1+ r_2$ <p>$\cos(\phi - \alpha)=\frac{r_1^2 - (r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}$</p> <p>$=-1$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/bdae2058-d0c1-45c4-d701-2872f2a2cb00/public"></p> </div>

##### <Success>Two circles intersect each other at two points</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> $\cos(\phi - \alpha) = \frac{r_1^2 - (r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}=1$ <p>$d_{o_1,o_2} = |r_1 - r_2|$</p> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2d4b4f49-03e4-43ed-5da5-c5b292cfb600/public"></p> </div> <div style='float: right; width:25%; hight:50%'> <!----> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/35c08fe1-70d5-4d94-2f41-e6cc51166e00/public"> <img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/6ff02c16-7d13-4a59-8d8d-0d924a8e3500/public"></p> </div>

### <Success>Notes</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$ \left|\frac{r_1^2 - (r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}\right|<1$ (two circles intersect at exactly 2 points)</p> <p>$ \left|\frac{r_1^2 - (r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}\right|=1$ (two circles touch each other)</p> <p>$ \left|\frac{r_1^2 - (r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}\right|>1$ (two circles donot intersect)</p> <p>$\ $<br> $|r_1 - r_2| < d_{o_1,o_2} < r_1 + r_2$</p> </div> <div style='float: right; width:0%;'> <p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-12-chapter-11-circles-l-11_13-fdjwbn2dkpm/img/320-2655.6.jpg"></p> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$cos \alpha = \frac{(c-a)}{d_{o_1,o_2}}=1$</p> <p>$\sin \alpha = 0 \Rightarrow \alpha = 0$</p> <p>$x_1 = c + r_2 \cos \phi_1$</p> <p>$y_1 = d + r_2 \sin \phi_1$</p> <p>$\cos(\phi - \alpha) = \frac{r_1^2-(r_2^2 + d_{o_1,o_2}^2)}{2r_2 d_{o_1,o_2}}$</p> <p>$=\frac{3^2 - (3^2 + 5^2)}{2 \times 3 \times 5} = -\frac{5}{6}$</p> </div> <p>======</p> <h3 id="successexamplesuccess"><Success>Example</Success></h3> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>$x_2 = c + r_2 \cos \phi_2$</p> <p>$y_2 = d + r_2 \sin \phi_2$</p> <p>$\cos \phi = -\frac{5}{6}$</p> <p>$\phi_1 = \cos^{-1}[\frac{-5}{6}] \in [0, \pi]$</p> <p>$\phi_2 = 2\pi - \cos^{-1} (-5/6)$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ee8010bd-4fba-4c62-030a-31de3d2b6000/public"></p> </div>

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