Question

find the equation of an ellipse whose foci are (-3,5) and (5,5) and major axis is 10​

Answer 1

the equation of an ellipse whose foci are (-3,5) and (5,5) and major axis is 10​ is  $\dfrac{(x-1)^2}{25} + \dfrac{(y-5)^2}{9}$

• Given,
• 2a = 10 ⇒ ∴ a = 5
• standard form equation of ellipse is:
• $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$
• foci (-3,5) and (5,5)
• mid-point is given by,  $\dfrac{-3+5}{2}, \dfrac{5+5}{2} = (1, 5)$
• ∴ (1, 5) is the mid-point.
• Now, x = X+1, y = Y+5
• ⇒ x = ae + 1, y = 0+5
• ⇒ (ae+1, 5)
• ae+1 = 5
• ae = 4
• e = 4/a = 4/5
• ∴ e = 4/5
• e^2 = 1 - b^2/a^2
• 4^2/5^2 = 1- b^2/5^2
• 4^2 = 5^2-b^2
• b^2 = 3^2
• ∴ b =3
• Therefore the equation of an ellipse is given by,
• $\dfrac{(x-1)^2}{25} + \dfrac{(y-5)^2}{9}$