Circle Summary Engineering maths

Circles
Circle
Circle is defined as the locus of a point which moves in a plane such
that its distance from a fixed point in that plane is constant.
P (x, y)
r

C (0, 0)



The fixed point is called the centre and the constant distance is called
the radius.

Standard Equation of a Circle
Equation of circle having centre ( h , k) and radius a is
( x − h )2 + ( y − k)2 = a 2. This is also known as central form of equation
of a circle.
Some Particular Cases of the Central Form
(i) When centre is ( 0, 0), then equation of circle is x 2 + y 2 = a 2.
Y
a


X
O a




(ii) When the circle passes through the origin, then equation of the
circle is x 2 + y 2 − 2hx − 2ky = 0.
Y


C(h, k)
a
k
O h M X

, (iii) When the circle touches the X-axis, the equation is
x 2 + y 2 − 2hx − 2ay + h 2 = 0.
Y



C (h, k)
a

O X
M
(iv) Equation of the circle, touches the Y-axis is
x 2 + y 2 − 2ax − 2ky + k2 = 0.
Y



C (h, k)
M a


X
O

(v) Equation of the circle, touching both axes is
x 2 + y 2 − 2ax − 2ay + a 2 = 0.
Y




M a C (a, a)

a
X
O


(vi) Equation of the circle passing through the origin and centre
lying on the X-axis is x 2 + y 2 − 2ax = 0.
Y




X' a
O X
C (a, 0)



Y'