Question

If equation of circle at centre ( acos alpha ,Asin alpha) with radius a is

Answer 1

Equation of a circle

If ( h, k ) are the coordinates of the centre of circle with radius ' r ' units the the Equation of circle would be

$\quad \quad \bf {(x - h)}^{2} + {(y - k)}^{2} = {r}^{2}$

Given :

Coordinates of the centre of the circle ( h, k) = ( a cos α, a sin α )

• h = a cos α
• k = a sin α

Radius of the circe ( r ) = a units

Substitute the values in formula

$\sf \Rightarrow {(x - acos \alpha )}^{2} + {(y - asin \alpha )}^{2} = {a}^{2}$

$\sf \Rightarrow {x}^{2} + {a}^{2} cos^{2} \alpha -2(x)(acos \alpha ) + {y}^{2} + {a}^{2}sin ^{2} \alpha - 2(x)(asin \alpha ) = {a}^{2}$

$\sf \Rightarrow {x}^{2} + {a}^{2} (cos^{2} \alpha + sin^{2} \alpha )-2xacos \alpha+ {y}^{2} - 2xasin \alpha = {a}^{2}$

$\sf \Rightarrow {x}^{2} + {a}^{2} -2xacos \alpha+ {y}^{2} - 2xasin \alpha = {a}^{2}$

$\sf \Rightarrow {x}^{2} + y^2 = 2xacos \alpha+ 2xasin \alpha$