Circle : Introduction

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.

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 circle

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.

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 xaxis  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 yaxis  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