Content-
Definition :
Circle : it is the locus of a point that in which moves in a plane such that its distance from a fixed point in a plane is always constant. The fixed point is known as centre and the constant distance is known as radius of the circle.
Equation of circle—
According to definition –
Distance between points C( h , k) and P( x, y ) is equal to the radius of the circle.
Centre : C( h , k)
Radius : r
Equation of a Circle :
( x – h )2 + ( y – k )2 = r2
If centre is origin —
Centre : O ( 0 , 0 )
Radius : r
( x – 0 )2 + ( y – 0 )2 = r2
x 2 + y2 = r2
Equation of a circle in different situations :
Case 1 : Centre ( h , k ) and passes through the origin(0 , 0)
r = √(h – 0) 2 + (k – 0)2 = h 2 + k2
r = √(h 2 + k2 )
( x – h )2 + ( y – k )2 = h 2 + k2
x 2 + y2 — 2.h.x — 2.k.y = 0
Case 2 : Centre ( h , k ) and touches the x-axis.
According to figure – it is clear that radius will be equal to k.
( x – h )2 + ( y – k )2 = k2
Intersection with x—axis :
And this equation touches x—axis i.e. y = 0
( x – h )2 + ( 0 – k )2 = k2
x 2 — 2.h.x + h2 = 0.
Or
( x – h )2 = 0
Case 3 : Centre ( h , k ) and touches the y-axis.
According to figure – it is clear that radius will be equal to h.
( x – h )2 + ( y – k )2 = h2
Intersection with y—axis :
And this equation touches y—axis i.e. x = 0
( 0 – h )2 + ( y – k )2 = h2
y 2 — 2.k.y + k2 = 0. Or
( y – k )2 = 0
Case 4 : When circle touches the both axis.
According to figure – it is clear that centre will be ( h , h ) radius will be equal to h.
( x – h )2 + ( y – h )2 = h2
x2 + y2 — 2.h.x — 2.k.y + h2 = 0
But the centre of the circle could be any of the four quadrants. Then co-ordinates of the centre taken as
( ± h , ± h ).
Then equation of the circle is —
x2 + y2 ± 2.h.x ± 2.k.y + h2 = 0