### <Success>Circles: Lecture-2</Success> <div style='float: left; width:50%;'> </div>

#### <Success>EQUATION OF A CIRCLE PASSING THROUGH THREE NON - COLLINEAR POINTS </Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <ul> <li> <p>$(x_1 y_1)$</p> </li> <li> <p>$(x_2 y_2)$</p> </li> <li> <p>$(x_3 y_3)$</p> </li> </ul> </div> <div style='float: right; width:1%;'> <!----> </div>

#### <Success>Equation of a circle passing through three non - collinear points </Success> <div style='float: left; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/47ef3489-20a4-4d09-237b-e99632821300/public"></p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2b917875-5f63-474e-9022-99014ed91000/public"></p> <!----> </div>

#### <Success>Equation of a circle passing through three non - collinear points </Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>$(\frac{y - \frac{(y_1 + y_2)}{2}}{x-\frac{(x_1 + x_2)}{2}}) \times \frac{(y_2 - y_1)}{(x_2 - x_1)}$</p> <p>$=-1$</p> <p>$r=\sqrt{(x_0 - x_2)^2 + (y_0 - y_2)^2}$</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9eaf2a65-a89d-4423-2d77-50d4a68ae600/public"></p> <!----> </div>

#### <Success>Equation of a circle passing through three non - collinear points </Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> B1: $(y_0 - \frac{(y_1 + y_2)}{2}) (y_2 - y_1) + (x_0 - \frac{(x_1 + x_2)}{2}) \times (x_2 - x_1) = 0$ <p>B2: $(y_0 - \frac{(y_2 + y_3)}{2}) (y_3 - y_2) + (x_0 - \frac{(x_2 + x_3)}{2}) \times (x_3 - x_2)=0$</p> <p>$(x_0, y_0)$ is the centre of the circle which passes through $p_1$, $p_2$ and $p_3$.</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/df2b0571-8429-4ddc-7432-03da0cedf200/public"></p> </div>

#### <Success>Equation of a circle passing through three non - collinear points </Success> <div style='float: left; width:0%;'> </div> <div style='float: right; width:60%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/aff3c576-d43f-4366-bff5-6c042ee81200/public"></p> <!----> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> Show that the four points (1, 0), (2, -7), (8, 1) and (9, -6) are con cyclic. <p><strong>Solution:</strong> $(\frac{y + 3.5}{x - 1.5}) \times -7 = -1$</p> <p>$y + 3.5 = \frac{x}{7} - \frac{1.5}{7}$</p> <p>$y - 0.5 = -7x + 30.5$</p> <p>$(\frac{y - 0.5}{x - 4.5}) \times \frac{1}{7} = -1$</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5fabcdbf-119d-4f10-a22c-6ed056290100/public"></p> <!----> </div>

### <Success>Solution</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> $ y_0 + 3.5 = \frac{x_0}{7} - \frac{1.5}{7}$ <p>$ y_0 - 0.5 = -7 x_0 + 31.5$</p> <p>$ \frac{x_0}{7} - \frac{1.5}{7} - 3.5 = -7 x_0 + 32$</p> <p>$ \frac{50 x_0}{7} = \frac{224 + 1.5 + 24.5}{7}$</p> <p>$ =\frac{250}{7} $</p> <p>$x_0 = 5$</p> <p>$\therefore y_0 =-7 x_0 + 32 $</p> <p>$ =-7 \times 5 + 32 $</p> <p>$ =-3 $</p> <p>$\therefore$ the centre of the circle is at (5, -3).</p> </div> <div style='float: right; width:30%;'> <!----> <!----> </div>

### <Success>Solution</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> radius $r = OP_1 =\sqrt{(5-1)^2+(-3)^2} =5$ <p>$(x-x_0)^2 + (y-y_0)^2 = r^2$</p> <p>i.e., $(x-5)^2+(y+3)^2=25$</p> <p>Check to see if (9, -6) lies on this circle.</p> <p>$(9-5)^2 + (-6+3)^2 = 16 + 9 = 25$</p> </div> <div style='float: right; width:30%;'> <!----> <!----> </div>

### <Success>General form of circle</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> $P_1(x_1, y_1), P_2(x_2, y_2)$ and $P_3(x_3, y_3)$ <p>GENERAL FORM $C: x^2 + y^2 + 2gx + 2fy + c = 0$</p> <p>since $P_1$ lies on C $\Rightarrow x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c = 0$</p> <p>$x_2^2 + y_2^2 + 2gx_2 + 2fy_2 + c = 0$</p> <p>$x_3^2 + y_3^2 + 2gx_3 + 2fy_3 + c = 0$</p> </div> <div style='float: right; width:0%;'> <!----> </div>

### <Success>General form of circle</Success> <div style='float: left; text-align:left; width:70%;font-size:30px;'> <p>$ \frac{(y_2-y)}{(x_2-x)} \times \frac{(y_1-y)}{(x_1-x)}$</p> <p>$=-1$</p> <p>$ (x-x_1) (x-x_2) + (y-y_1)(y-y_2)= 0$</p> <p>$ x^2 + y^2 - (x_1 + x_2) x - (y_1 + y_2) y + (x_1 x_2 + y_1 y_2) = 0$</p> <p>$x^2 + y^2 + 2gx + 2fy + c = 0$</p> </div> <div style='float: right; width:30%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e2196037-d2eb-4edc-d3d2-9b8afba39f00/public"></p> <!----> </div>

### <Success>Position of a point w.r.t. a circle </Success> <div style='float: left; text-align:left; width:60%;font-size:30px;'> <p>P(a, b)</p> <p>$C:x^2 + y^2 + 2gx + 2fy + c = 0$</p> <p>$r = \sqrt{g^2 + f^2 - c}$</p> <p>P is inside the circle if and only if</p> <p>$\sqrt{(a+g)^2 + (b+f)^2}<\sqrt{g^2 + f^2 - c}$</p> <p>$a^2 + b^2 + 2ag + 2bf + c < 0$</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/174e91b1-3ac9-49e0-d963-afecc11fe600/public"></p> <!----> </div>

### <Success>Position of a point w.r.t. a circle </Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> Given a point P(a, b) and a circle, <p>$C:x^2 + y^2 + 2gx + 2fy + c = 0$</p> <p>$a^2 + b^2 + 2ga + 2fb + c$</p> <p>$a^2 + b^2 + 2ga + 2fb + c = 0 \Rightarrow$ P lies on the circle C</p> <p>$a^2 + b^2 + 2ga + 2fb + c < 0 \Rightarrow$ P lies inside C</p> <p>$a^2 + b^2 + 2ga + 2fb + c > 0 \Rightarrow$ P lies outside C</p> </div> <div style='float: right; width:1%;'> <!----> </div>

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