Find the equation of an ellipse whose eccentricity is 2/3, the latus - rectum is 5 and the centre is at the origin.
Given that we need to find the equation of the ellipse whose eccentricity is , latus - rectum is 5 and centre is at origin.
Let us assume the equation of the ellipse is - - - - (1) (a2>b2) since centre is at origin.
We know that eccentricity of the ellipse is
⇒
⇒
⇒ 9(a2 - b2) = 4a2
⇒ 5a2 = 9b2
⇒ ..... - - - (2)
We know that length of the latus - rectum is
⇒
⇒
⇒
⇒
⇒
From (2),
⇒
⇒
The equation of the ellipse is
⇒
⇒
⇒
⇒ 20x2 + 36y2 = 405
∴ The equation of the ellipse is 20x2 + 36y2 = 405.