Question

given the ellipse

\\\frac{x^2}{25} + \frac{y^2}{9} = 1\\

find both foci.

enter your answer as ordered pairs separated by commas. for example, if the foci are at \\((0,4)\\) and \\((0,-4)\\) your answer will be \(0,4),(0,-4)\ round to three decimal places.

Explanation:

Step1: Identify ellipse parameters

Identify \(a^2\) and \(b^2\) from the equation.

$$a^2 = 25, \quad b^2 = 9$$

Step2: Calculate focal distance

Use the formula \(c^2 = a^2 - b^2\).

$$c^2 = 25 - 9 = 16 \implies c = 4$$

Step3: Determine foci coordinates

Since \(a^2\) is under \(x^2\), the foci lie on the x-axis.

$$(4,0), (-4,0)$$

Answer:

(4,0),(-4,0)