Question

The foci of the ellipse \(\frac{x^2}{49} + \frac{y^2}{24} = 1\) are:

• A. \( (7,0) \) and \( (-7,0) \)
• B. \( (6,0) \) and \( (-6,0) \)
• C. \( (4,0) \) and \( (-4,0) \)
• D. \( (5,0) \) and \( (-5,0) \)
• E. \( (3,0) \) and \( (-3,0) \)

Hint:

To find the foci of an ellipse, identify \(a^2\) and \(b^2\) (the coefficients under \(x^2\) and \(y^2\)), determine which is larger to find the direction of the major axis, and use \(c = \sqrt{a^2 - b^2}\) to locate the foci along the major axis.

Answer 1

The Correct Option is D

For an ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the distance of each focus from the center along the major axis is \(c\), where \(c^2 = a^2 - b^2\).
Here, the major axis is along the x-axis (since \(a^2 = 49\) is greater than \(b^2 = 24\)), so \(a = 7\) and \(b = \sqrt{24}\).
Calculate \(c\): \[ c = \sqrt{a^2 - b^2} = \sqrt{49 - 24} = \sqrt{25} = 5 \] Thus, the foci of the ellipse are located at \((\pm c, 0)\), or: \[ (5, 0) { and } (-5, 0) \]