Question

Write an equation for an ellipse centered at the origin, which has foci at (\pm8,0)(±8,0)left parenthesis, plus minus, 8, comma, 0, right parenthesis and vertices at (\pm17,0)(±17,0)left parenthesis, plus minus, 17, comma, 0, right parenthesis

Answer 1

The equation of the ellipse is
`(x^(2))/((17)^(2))+(y^(2))/((15)^(2))=1`

Explanation:

The standard form of the equation of an ellipse with center (0 , 0) is

`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` , where

• The coordinates of the vertices are (± a , 0)

• The coordinates of the foci are (± c , 0) , where c ² = a² - b²

∵ The ellipse is centered at the origin

∴ The center of it is (0 , 0)

∵ It has foci at (± 8 , 0)

- The coordinates of the foci are (± c , 0)

∴ c = ±8

∵ It has Vertices at (± 17 , 0)

- The coordinates of the vertices are (± a , 0)

∴ a = ±17

∵ c² = a² - b²

- Substitute the values of c and a to find b

∴ (8)² = (17)² - b²

∴ 64 = 289 - b²

- Add b² to both sides

∴ b² + 64 = 289

- Subtract 64 from both sides

∴ b² = 225

- Take √ for both sides

∴ b = ± 15

∵ The equation of the ellipse is
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`

- Substitute the values of a and b in it

∴
`(x^(2))/((17)^(2))+(y^(2))/((15)^(2))=1` ⇒
`(x^(2))/(289)+(y^(2))/(225)=1`

∴ The equation of the ellipse is
`(x^(2))/((17)^(2))+(y^(2))/((15)^(2))=1`