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