Question

If the tangent to the ellipse x² + 4y² = 16 at the point P (θ) is a normal to the circle x² + y² – 8x – 4y = 0 then θ equals

(a)  π/2

(b)  π/4

(c)  0

(d)  -π/4 

Answer 1

Correct option  (a) , (c)

Explanation:

Given ellipse is  x²/16 + y²/4 = 1

Equation of tangent at P (θ) is P(acosθ ,bsinθ) i.e. P (4cosθ , 2sinθ) is

4x cosθ/16 + 2ysin θ/4 ....(1)

xcosθ  + 2ysinθ = 4  .......(1)

(1) is a normal to the circle x² + y² – 8x – 4y = 

The equation (1) passes through the centre (4,2) of the circle. 

4cosθ + 4sinθ = 4

cosθ  + sinθ = 1

squaring

1 + sin2 = 1

sin2θ = 0

2θ = 0 or π

Hence, θ = 0 or π