If S and S’ are two foci of the ellipse and B is an end of the minor axis such that Δ BSS’ is equilateral, then write the eccentricity of the ellipse.

Question

Philoid · Accepted Answer

Given that S and S’ are the foci of the ellipse .

It is told that ΔBSS’ is equilateral, where B is the end of the minor axis. We need to find the eccentricity of the ellipse.

Let us assume that B = (0,b)

We know that foci of the ellipse are (±ae,0).

We know that the distance between the foci is 2ae.

Let us find the distance SB

We know that the distance between the points (x₁,y₁) and (x₂,y₂) is .

⇒

⇒ .

We know that sides of an equilateral triangle are equal.

⇒ SB = 2ae

⇒ SB² = 4a²e²

⇒ a²e² + b² = 4a²e²

We know that b² = a²(1 - e²),

⇒ a²e² + a² - a²e² = 4a²e²

⇒ a² = 4a²e²

⇒

⇒ .