Question

The eccentric angles of two points P and Q of the ellipse 4x² + y² = 4 differ by 2π/3. Show that the locus of the point of intersection of the tangents at P and Q is the ellipse 4x² + y² = 16.

Answer 1

Given equation of the ellipse is 4x² + y² = 4

\(\frac{x^2}{1} + \frac {y^2}{4} = 1\)

Let P(θ₁) and Q(θ₂) be any two points on the given ellipse such that

θ₁ – θ₂ = 2π/3

The equation of the tangent at point P(θ₁) is

……(i)

The equation of the tangent at point Q(θ₂) is

\(\frac {x \,cos \,θ_2}{1} + \frac {y\, sin\, θ_2}{2} = 1\)

Multiplying equation (i) by cos θ₂ and equation (ii) by cos θ₁ and subtracting, we get

4x² + y² = 16, which is the required equation of locus.