Question

The angle between the pair of tangents from a point P to the circle S ≡ x² + y² + 4x − 6y + 9 + 4cos² α = 0 is 2α. Show that the point P lies on the circle x² + y² + 4x − 6y + 9 = 0 and hence find the equation of the director circle of S = 0

Answer 1

The centre of the circle S = 0 is (−2, 3) and its radius is 2sin α (note that 2α being the angle between the tangents, we have 0 < α < π/2). Let P = (h, k) as shown in Fig. Then

Therefore

(h + 2)² + (k - 3)² = 4

Hence, (h, k) lies on the circle

(x + 2)² + (y − 3)² = 4 or x² + y² + 4x − 6y + 9 = 0

If α =  π/4, then 2α =  π/2 and therefore the equation of the director circle of S ≡ x² + y² + 4x − 6y + 11 = 0 is given by x² + y² + 4x − 6y + 9 = 0.