SHORT NOTES: Circle

Equation of a circle in various forms:
General equation of a circle:

$x^2+y^2+2gx+2fy+c=0$

centre is $\left ( -g,-f \right )$

radius is $\sqrt{g^2+f^2-c}$

(i)circle with centre at $\left ( h,k \right )$ and radius $r$ is:

$\left ( x-h \right )^2+\left ( y-k \right )^2=r^2$

(ii)circle with end points of diameter are given as $\left ( x_1,y_1 \right )$ and $\left ( x_2,y_2 \right )$ :

$\left ( x-x_1 \right )\left ( x-x_2 \right )+\left ( y-y_1 \right )\left ( y-y_2 \right )=0$

(iii)circle passing through three points $\left ( x_1,y_1 \right )$ , $\left ( x_2,y_2 \right )$ , $\left ( x_3,y_3 \right )$ :

$\begin{vmatrix} x^2+y^2 &x &y &1 \\ x_1^2+y_1^2 & x_1& y_1& 1\\ x_2^2+y_2^2& x_2& y_2& 1\\ x_3^2+y_3^2& x_3& y_3& 1 \end{vmatrix}=0$

Equations of tangent:

(i)Slope form: the equation of a tangent of slope m to the circle $x^2+y^2=r^2$ is

$y=mx \pm r\sqrt{{1+m^2}}$ i.e $c= \pm r\sqrt{{1+m^2}}$

The coordinates of points of contact

$\left ( \pm \frac{rm}{\sqrt{{1+m^2}}},\mp \frac{r}{\sqrt{{1+m^2}}} \right )$

(ii)Point form: the equation of the tangent at point $\left ( x_1,y_1 \right )$ to circle

$x^2+y^2+2gx+2fy+c=0$ is

$xx_1+yy_1+g\left ( x+x_1 \right )+f\left ( y+y_1 \right )+c=0$

Pair of tangents from point $\left ( x_1,y_1 \right )$ to $x^2+y^2=a^2$ :

$\left ( x^2+y^2-a^2 \right )\left ( x_1^2+y_1^2-a^2 \right )=\left ( xx_1+yy_1-a^2 \right )^2$

Length of tangents from $\left ( x_1,y_1 \right )$ to $x^2+y^2+2gx+2fy+c=0$ :

$\sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c}$

Normal to a circle:

The normal at any point is a straight line perpendicular to the tangent at that point.

$m_1m_2=-1$

chord of contact:

the equation of the chord of contact of tangents drawn from a point P $\left ( x_1,y_1 \right )$ to the circle $x^2+y^2=a^2$ is

$xx_1+yy_1=a^2$

Parametric equations of a circle:

$x=a+r\cos \theta \ y=b+r\sin \theta$

is the required parametric equation of the circle

$\left ( x-a \right )^2+\left ( y-b \right )^2=r^2$

intersection of a circle with a straight line:

Let the circle $x^2+y^2=r^2$ and the line $y=mx+c$ substituting value of y

$x^2\left ( 1+m^2 \right )+2mcx+\left ( c^2-r^2 \right )=0$

$r> \left | \frac{c}{\sqrt{{1+m^2}}} \right |$ real and distinct point of intersection

$r= \left | \frac{c}{\sqrt{{1+m^2}}} \right |$ coincident point of intersection

$r< \left | \frac{c}{\sqrt{{1+m^2}}} \right |$ imaginary point of intersection

Angle of intersection of two circles:

$\cos \theta=\frac{r_1^2+r_2^2-d^2}{2r_1r_2}$

circles passing through the intersection point of a given circle and a given line:

$S:{x^2} + {y^2} + 2gx + 2fy + c = 0$

$L:px + qy + r = 0$

$F:S + \lambda L = 0$

$\Rightarrow \,\,\,\, {x^2} + {y^2} + (2g + \lambda p)x + (2f + \lambda q)y + c + \lambda r = 0$

equation of a circle through the points of intersection of two circles :

${S_1}:{x^2} + {y^2} + 2{g_1}x + 2{f_1}y + {c_1} = 0$

${S_2}:{x^2} + {y^2} + 2{g_2}x + 2{f_2}y + {c_2} = 0$

${F:{S_1} + \lambda {S_2} = 0}$

ELECTRICAL ENGINEERING
ENGINEERING MATHEMATICS	Electric Circuits and Fields
Signals and Systems	Electrical Machines
Power Systems	Control Systems
Electrical and Electronic Measurements	Power Electronics and Drives
Analog Electronics	Digital Electronics

Circle

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