Question

The angle between a pair of tangents drawn from a point P to the circle x² + y² + 4x – 6y + 9 sin²a + 13 cos²a = 0 is 2a.

The equation of the locus of the point P is –

• A. x² + y² + 4x – 6y + 4 = 0
• B. x² + y² + 4x – 6y – 9 = 0
• C. x² + y² + 4x – 6y – 4 = 0
• D. x² + y² + 4x – 6y + 9 = 0

Answer 1

Answer: (D)

x² + y² + 4x – 6y + 9 = 0

Answer 2

The centre (–2, 3) and the radius is

$\sqrt { 4 + 9 - 9 \sin ^ { 2 } \alpha - 13 \cos ^ { 2 } \alpha }$ = 2 sin a

Let (h, k) be a point from which tangents PA and PB are drawn to the given circle. Then as given CPA = ŠCPB = a

Also, sin a = $\frac { \mathrm { AC } } { \mathrm { PC } }$ = $\frac { 2 \sin \alpha } { P C }$

Ž PC² = 4 (Q AC = radius = 2 sin a)

Ž (h + k)² + (k – 3)² = 4

or locus is x² + y² + 4x – 6y + 9 = 0 on replacing h and k by x and y.