Bahan Ajar dan Bank Soal Matematika: Equation of Circle

Definiton of Circle

The circle is the locus of points on a flat, which has a distance equal to a certain point. Furthermore, the same distance is called the radius (r), while a certain point called the center (P) of the circle.

Equation of a circle with center at O(0, 0) with radius r :
$x^{2}+y^{2}=r^{2}$
Equation of a circle with center at P(α, β) with radius r : $\left ( x-\alpha \right )^{2}+\left ( y-\beta \right )^{2}=r^{2}$

The general form equation of a circle:

$x^{2} + y^{2} + Ax + By + C = 0$

where

$A=-2\alpha , B=-2\beta , C=\alpha ^{2}+\beta ^{2}-r^{2}$

so that it can also state that the circle has a center C with radius r:

$P\left ( -\frac{1}{2}A,-\frac{1}{2}B \right )$

$r=\frac{1}{2}\sqrt{A^{2}+B^{2}-4C}$

Equation of a tangent to a circle centered at O(0, 0) with radius r at the point $A\left ( x_{1},y_{1} \right )$

$x_{1}x + y_{1}y = r^{2}$

Equation of a tangent to a circle centered at O(0,0) with radius r, if the gradient = m $y = mx \pm \sqrt{m^{2}+1}$

Equation of a tangent to a circle centered at P(α, β) with radius r at the point $A\left ( x_{1},y_{1} \right )$ is : $\left ( x-\alpha \right )\left ( x_{1}-\alpha \right ) +\left ( y-\beta \right )\left ( y_{1}-\beta \right )=r^{2}$

Equation of a tangent to a circle centered at P(α, β) with radius r, if the gradient = m is $y-\beta =m\left ( x-\alpha \right )\pm \sqrt{m^{2}+1}$