JEE Advanced Model Paper 9 with solutions for online practice

The equation of the tangent at (3cosθ , 2sinθ) on

x2

9

+

y2

4

= 1 is

x

3

cosθ +

y

2

sinθ = 1 ..... (i)
The equation of director circle is x2+y2 = 9 + 4 = 13 ..... (ii)
The combined equation of CP and CQ is obtained by homogenizing equation (ii) with (i).
Thus combined equation is x2+y2 = 13 (

x

3

cosθ+

y

2

sinθ)2
⇒ (

13

9

cos2θ−1)x2 +

13

3

sinθ cosθ xy + (

13

4

sin2θ−1)y2 = 0
∴ Product of the slopes of CP and CQ
=

coefficient‌of‌x2

coefficient‌of‌y2

=

13 9 cos2θ−9

13 4 sin2θ−1

=

13cos2θ−9

13sin2θ−4

×

4

9

=

13cos2θ−9

13−13cos2θ−4

×

4

9

= −

4

9