Question

Find the equations of the circles which have the radius

root13 and which touch the line 2x−3y+1=0 at (1,1)

Answer 1

Given that, the circle have the radius

13

and touches the line 2x−3y+1=0 at (1,1).

Point circle at (1,1) is (x−1)

2

+(y−1)

2

=0

i.e S:x

2

+y

2

−2x−2y+2=0

∴ Equation of the circle is of the form S+λP=0 where P is 2x−3y+1=0

⇒(x

2

+y

2

−2x−2y+2)+λ(2x−3y+1)=0

⇒x

2

+y

2

+(2λ−2)x−(3λ+2)y+(λ+2)=0

Radius of above circle is

13

.

⇒(λ−1)

2

+(

2

3λ+2

)

2

−(λ+2)=13

⇒4(λ−1)

2

+(3λ+2)

2

−4(λ+2)=52

⇒13λ

2

=52

⇒λ=±2

∴ Required equations of circles are x

2

+y

2

+2x−8y+4=0 and x

2

+y

2

−6x+4y=0

Hence, option C.