Find the standard equation for the ellipse, using the given characteristic or characteristics. vertices: (0,+-7) foci: (0,+-√33)

The standard equation is x²/16 + y²/49 = 1

Step-by-step explanation:

* Lets revise the standard equation of the ellipse

- The standard form of the equation of an ellipse with  

   center (0 , 0) is x²/b² + y²/a² = 1 , where  

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

* The coordinates of the foci are (0 , ± c), where c² = a² - b²  

* Now lets solve the problem

∵ The vertices of the ellipse are (0 , -7) , (0 , 7)

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

∴ a = 7 , -7

∵ The coordinates of the foci are (0 , -√33) , (0 , √33)

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

∴ c = -√33 , √33

∵ c² = a² - b²

∵ a² = (7)² = 49

∵ c² = (√33)² = 33

∴ 33 = 49 - b² ⇒ subtract 49 from both sides

∴ -16 = -b² ⇒ multiply both sides by -1

∴ b² = 16

∵ The standard equation of the ellipse is x²/b² + y²/a² = 1

∴ The standard equation is x²/16 + y²/49 = 1