If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

Solution

Maths sir · 3 steps

Turn the sentence into 2b = 1/4(2ae), then use b² = a²(1-e²) to remove b.

1. Name the lengths. For x²/a²+y²/b²=1 with a > b, the minor axis is 2b and the foci (± ae, 0) are 2ae apart.
2. Write the condition. 2b = 1/4(2ae), so b = ae/4. Squaring gives b² = a²e²/16.

   a²(1-e²) = a²e²/16 ⇒ 16 - 16e² = e² ⇒ e² = 16/17

3. So. e = 4/√17, which is option (c). It is less than 1, as the eccentricity of an ellipse must be.

The idea to keep. Any ratio of ellipse lengths becomes an equation in e once you write each length in a, b, e and replace b² with a²(1-e²).

Answer from the NTA final answer key. Solution by PrepWiser.