What is the equation of the ellipse having foci (±2, 0) and the eccentricity \(\frac{1}{4}?\)

Concept:

Standard equation of an ellipse: \(\frac{{{\rm{\;}}{{\bf{x}}^2}}}{{{{\bf{a}}^2}}} + \frac{{{{\bf{y}}^2}}}{{{{\bf{b}}^2}}} = 1\) (a > b)

• Coordinates of foci = (± ae, 0)
• Eccentricity (e) = \(\sqrt {1 - {\rm{\;}}\frac{{{{\rm{b}}^2}}}{{{{\rm{a}}^2}}}} \) ⇔ a²e² = a² – b²

 
Calculation:

Given: Focus: (±2, 0) and eccentricity = 1/4

⇒ Focus = (±ae, 0) = (±2, 0)

⇒ ae = 2

⇒ a × (1/4) = 2

∴ a = 8

⇒ a² = 64

Now,

a²e² = a² – b²

⇒ 2² = 8² - b²

∴ b² = 60

Equation of an ellipse = \(\frac{{{\rm{\;}}{{\rm{x}}^2}}}{{64}} + \frac{{{{\rm{y}}^2}}}{{60}} = 1\)