Question

Equation of smallest circle passing through. the point of intersection of circle x² + y² – 6x+ 2y – 6 = 0 and line

x + y + 2 = 0 is –

• A. x² + y² – 2x + 6y + 2 = 0
• B. x² + y² + 2x – 6y + 2 = 0
• C. x² + y² + 4x + 4y – 4 = 0
• D. None

Answer 1

Answer: (A)

x² + y² – 2x + 6y + 2 = 0

Answer 2

Let circle is S ŗ x² + y² – 6x + 2y – 6 = 0

Line is L ŗ x + y + 2 = 0

Equation of required circle S + lL = 0

Ž x² + y² – 6x + 2y – 6 + l(x + y + 2) = 0

Ž x² + y² + (l – 6)x + (l +2)y – 6 +2l = 0 .....(i)

Centre $\left( –\frac{(\lambda –6)}{2},–\frac{(\lambda + 2)}{2} \right)$

Circle will be smallest if L is diameter of (i) i.e. $\left( –\frac{(\lambda –6)}{2},\frac{–(\lambda + 2)}{2} \right)$ lies on x + y + 2 = 0