Circle

A circle is a locus of a point which moves along a path such that the distance from a fixed point is always a constant. The fixed point is known as center and the fixed distance or constant is known as radius of the circle.

Equation of circle with center (h, k) and radius r

The equation of circle with center (h, k) and radius r is given by

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

If the center is origin, i.e, (0, 0), then, equation of circle changes to

$x^2+y^2=r^2$

General equation of a circle

General equation of a circle is given by

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

Radius of the circle

Radius of a circle having equation $x^2+y^2+2gx+2fy+c=0$ is given by

$r=\sqrt{g^2+f^2-c}$

Note:

If $g^2+f^2-c>0$ , then the circle is real.

If $g^2+f^2-c=0$ , then the circle is a point circle.

If $g^2+f^2-c<0$ , then the circle is an imaginary circle.

Some important points:

For a circle

Coefficient of $x^2$ = Coefficient of $y^2$

Coefficient of xy term in the equation of the circle is equal to zero.

If two circles have same center, then they are concentric.

Concyclic quadrilateral

If all four vertices of a quadrilateral lie on a circle, then the quadrilateral is said to be a concyclic quadrilateral. The vertices of that quadrilateral are said to be concyclic.

Equation of circle on a given diameter

If (x1,y1) and (x2,y2) are two points on the diameter of a circle,

Then the equation of the circle is given by

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

Graph Universe

CIRCLE