Question

Obtain the equation of the circle orthogonal to both the circles x² + y² + 3x – 5y+ 6 = 0 and 4x² + 4y² – 28x +29 = 0 and whose centre lies on the line 3x + 4y + 1 = 0.

Answer 1

Given circles are x² + y² + 3x – 5y + 6 = 0 ...........(i) 

and 4x² + 4y² – 28x + 29 = 0 

Let the required circle be x² + y² + 2gx + 2fy + c = 0 ..........(iii) 

Since circle (iii) cuts circles (i) and (ii) orthogonally 

or 40g – 20f = – 5. ..........(vi) 

Given line is 3x + 4y = – 1 ..........(vii) 

Since centre (– g, – f) of circle (iii) lies on line (vii), 

∴ – 3g – 4g = – 1 .........(viii)