Question

Let a circle \( C \) touch the lines \( L_{1}: 4 x-3 y+K_{1}=0 \) and \( L_{2}: 4 x-3 y+K_{2}=0, K_{1}, K_{2} \in R \). If a line passing through the centre of the circle \( C \) intersects \( L_{1} \) at \( (-1,2) \) and \( L_{2} \) at \( (3,-6) \), then the equation of the circle \( C \) is :

Answer 1

Co-ordinate of centre C ≡ \((\frac{3+(-1)}{2},\frac{-6+2}{2})≡ ( 1 , − 2 ) \)

L₁ is passing through A

⇒ −4 − 6 + K₁ = 0

⇒ K₁ = 10

L₂ is passing through B

⇒ 12 + 18 + K₂ = 0

⇒ K₂ = −30

Equation of L₁: 4x−3y+10 = 0

Equation of L₂: 4x−3y−30 = 0

⇒ Radius = 4

Equation of circle (x−1)² + (y+2)² = 16