Questions: Analyze the equation. That is, find the center, vertices, and foci of the ellipse and graph it. 25 x^2+y^2=100 What are the coordinates of the center? (Type an ordered pair.)

Solution Steps

The given equation is: \[ 25x^2 + y^2 = 100 \]

Divide both sides by 100 to get: \[ \frac{25x^2}{100} + \frac{y^2}{100} = 1 \] \[ \frac{x^2}{4} + \frac{y^2}{100} = 1 \]

The standard form of an ellipse is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

Here, \( h = 0 \) and \( k = 0 \), so the center is at: \[ (0, 0) \]

For the ellipse \(\frac{x^2}{4} + \frac{y^2}{100} = 1\):

• \( a^2 = 4 \) so \( a = 2 \)
• \( b^2 = 100 \) so \( b = 10 \)

Vertices are at: \[ (0, \pm b) = (0, \pm 10) \]

The foci are found using \( c^2 = b^2 - a^2 \): \[ c^2 = 100 - 4 = 96 \] \[ c = \sqrt{96} \approx 9.7980 \]

Foci are at: \[ (0, \pm c) = (0, \pm 9.7980) \]

Final Answer