Question

Find the coordinates for the foci.

`(x^(2))/(16) + \frac{ {y}^(2) }{81} = 1`

Answer 1

(0, - √65) and (0, √65)

Explanation:

Standard Form Equation of an Ellipse

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

a - semi-major axis

b - semi-minor axis

The formula of a foci:

`F=√(a^2-b^2)\ if\ a>b\ \text{horizontal elipse}\\\\F=√(b^2-a^2)\ if\ a<b\ \text{vertical elipse}`

We have:

`(x^2)/(16)+(y^2)/(81)=1\\\\(x^2)/(4^2)+(y^2)/(9^2)=1`

a = 4, b = 9 → a < b → vertical

The foci:

`F=√(9^2-4^2)=√(81-16)=√(65)`

The coordinates of foci:

`(-F,\ 0)\ and\ (F,\ 0)\ if\ a>b\\\\(0,\ -F)\ and\ (0,\ F)\ if\ a<b`

Substitute:

`(0,\ -√(65))\ and\ (0,\ √(65))`