Conic Sections Question 3

The ellipse in the standard $(x,y)$ coordinate plane below is the graph of $\dfrac{(x-5)^2}{25} + \dfrac{(y-3)^2}{9} =1$ . Which of the following points are the foci of the ellipse?

(0,3)

and

(5,3)

(0,3)

and

(10,3)

(1,3)

and

(5,0)

(1,3)

and

(9,3)

(5,0)

and

(5,6)

Approach

Given an ellipse in standard form:

\frac{(x-h)^2}{a^2}+ \frac{(y-k)^2}{b^2} = 1

The foci are $c$ units away from the center along the major axis, where c is given by:

c^2=a^2-b^2

c^2 = 25-9

c^2 = 16

c = 4

The center is at $(5,3)$ . Moving 4 away in the horizontal direction:

(5\pm 4,3)

(1,3) \text{ and } (9,3)