### <Success> Circles: lecture-1</Success> <div style='float: left; width:50%;'> </div>

### <Success>Definition</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>Fixed distance of any point on the circle from O</p> <p>$=R \equiv Radius$</p> <p>Fixed point O</p> <p>$\equiv$ Centre of the circle.</p> <p>A chord passing through the centre is called a Diameter.</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9cbeec04-f8d4-4ca1-48ac-cd4125627600/public"></p> <!----> </div>

### <Success>Equation of straight line</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>slope $=\frac{2-1}{1-0}-1$</p> <p>$\frac{y-1}{x-0}=1$</p> <p>$y=x+1$</p> <p>$8 \neq 7$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/07161893-0e46-4588-4eb2-7145f6aef800/public"></p> <!----> </div>

### <Success>Equation of a circle</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>CENTRE : (h, K), \quad \quad RADIUS : R</p> <p>$ \triangle OPQ$</p> <p>Pythagoras theorem</p> <p>$(OP)^2 = (OQ)^2 + (PQ)^2$</p> <p>$R^2 = (x-h)^2 + (y-K)^2$</p> <p>CENTRE : (1, 2)</p> <p>RADIUS : 5 = R</p> <p>$(x,y)=(-1,3)$</p> <p>$25 \neq 4 + 1$</p> <p>CENTRE - RADIUS FORM</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/46dbafc9-75b5-4a18-98e8-53a5c53ae600/public"></p> <!----> </div>

### <Success>Equation of a circle</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> $(x-3)^2+(y-7)^2=64$ <p>CENTRE: $(3,7)$</p> <p>RADIUS: 8</p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/0a7319e8-7558-486a-af09-7d23d4f39b00/public"></p> </div>

#### <Success>Equation of a circle in parametric form </Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>Circle with centre (h, k) and Radius R</p> <p>$OP=R$</p> <p>$OQ=R \cos \theta$</p> <p>$QP=R \sin \theta$</p> <p>$x=h+R \cos \theta$</p> <p>$y = k + R \sin \theta$</p> <p>$(x-h)^2=R^2 \cos^2 \theta$</p> <p>$(y-k)^2=R^2 \sin^2 \theta$</p> <p>Circle: $\{(x, y) \mid x = h + R \cos \theta,\\ \quad \quad \quad \quad \quad\quad y = k + R \sin \theta\}$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/942678f2-265b-467d-d6f4-ac0b32bdbc00/public"></p> <!----> </div>

#### <Success>Equation of a circle in parametric form </Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>Circle = (x, y) | $x=h+R \cos \theta$,</p> <p>$\quad \quad \quad \quad y = k + R \sin \theta$</p> <p>$\theta \in (0, 2 \pi)$</p> <p><strong>1)</strong> If a point (x, y) belongs to a circle having centre (h, K) and radius R,</p> <p>than $x = h + R \cos \theta, y = k + R \sin \theta $</p> <p>for some $\theta \in (0,2 \pi)$</p> <p><strong>2)</strong> For any angle $\theta \in (0, 2 \pi)$ the points $(h + R \cos \theta, k + R \sin \theta)$ belongs to the circle having centre (h, K) and radius R.</p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/57c3745c-d8a0-41f0-e42e-9ec34486ed00/public"></p> </div>

### <Success>Equation of a circle in general form</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$(x-h)^2 + (y-k)^2 = R^2$</p> <p>$x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = R^2$</p> <p>$[x^2 + y^2 - 2hx - 2ky + ck^2 + h^2 - R^2] = 0$</p> <p>$\rightarrow x^2 + y^2 + 2gx + 2fy + c = 0$</p> <p>$ax^2 + by^2 + cxy + 2dx + 2ey + f = 0$</p> <p>$\rightarrow$ general $2^{nd}$ degree equation</p> <p>$a=b$ and $c=0$</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-1-circles-l-1_13-_w78ish61pm/img/365-3046.8.jpg"></p> </div>

### <Success>Equation of a circle in general form</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> $ a x^2 + a y^2 + 2dx+ 2ey + f = 0$ <p>$ x^2 + y^2 + \frac{2d}{a} x + \frac{2 e}{a} y+\frac{f}{a} = 0$</p> <p>$\rightarrow x^2 + y^2 + 2gx + 2fy + c = 0$</p> <p>$(x^2 + 2gx + g^2) + (y^2 + 2fy + f^2) + c - g^2 - f^2 = 0$</p> <p>$(x + g)^2 + (y + f)^2 = (g^2 + f^2 - c)$</p> <p>$(x - h)^2 + (y - k)^2 = R^2 \geq 0$</p> <p>$g^2 + f^2 - c \geq 0$</p> <p>$R = \sqrt{g^2 + f^2 - c}$</p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9e3c0e5d-0e45-4630-a8ba-7b208ff49300/public"></p> </div>

### <Success>Equation of a circle in general form</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> $x^2 + y^2 + 2gx + 2fy + c = 0$ is a circle if and only if <p>$g^2 + f^2 - c \geq 0$, in which case the radius of the circle is</p> <p>$R=\sqrt{g^2 + f^2 - c}$ and the centre of the circle is</p> <p>$(h, k) = (-g, -f)$</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-1-circles-l-1_13-_w78ish61pm/img/429-3446.4.jpg"></p> </div>

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