Question

The tangent and the normal at a point P on an ellipse `x^2/a^2 + y^2/b^2` = 1 meet its major axis in T and T' so that TT' = a then e²cos²θ + cosθ (where e is the eccentricity of the ellipse) is equal to ______.

Options

• 1

• `1/2`

• `2/3`

• `1/4`

Answer 1

The tangent and the normal at a point P on an ellipse `x^2/a^2 + y^2/b^2` = 1 meet its major axis in T and T' so that TT' = a then e²cos²θ + cosθ (where e is eccentricity of the ellipse) is equal to 1.

Explanation:

Let the eccentric angle of the point P be θ so that tangent is `(xcosθ)/a + (ysinθ)/b` = 1  ...(i)

and normal is `(ax)/cosθ - (by)/sinθ` = a² – b²  ...(ii)

These lines meet the major axis y = 0 at points T and T' such that TT' = a

So T is `(a/cosθ, 0)` and T' is `(((a^2 - b^2)cosθ)/a, 0)`

∴ TT' is `a/cosθ - ((a^2 - b^2)cosθ)/a` = a (given)

or a² – a²e²cos²θ = a²cosθ

or 1 – e²cos²θ = cosθ

or e²cos²θ + cosθ – 1 = 0

or e²cos²θ + cosθ = 1