Question

Prove that the equation x² + y² – 2x – 2λ y – 8=0, where λ is a parameter, represents a family of circles passing through two fixed points A and B on the x-axis. Also find the equation of that circle of the family, the tangents to which at A and B meet on the line x + 2y + 5 = 0. 

Answer 1

∴The equation is represents a family of circles passing through the intersection of S = 0 & L = 0.

∴ On solving (ii) and (iii), we get

∴ The fixed point are A (4, 0) and B (-2, 0) from the diagram, the perpendicular bisector of AB is con. Current with the tangents at P 

If P is the point of intersection of tangents then CB is perpendicular to BP

∴ Equation of the required circle is