Question

The coordinates of the foci of the ellipse \( 16x^2 + 9y^2 = 144 \) are

• A. \( (\pm 7, 0) \)
• B. \( (0, \pm \sqrt{7}) \)
• C. \( (\pm \sqrt{7}, 0) \)
• D. \( (0, \pm 7) \)

Hint:

Always check which denominator is larger to determine the direction of the major axis of an ellipse.

Answer 1

The Correct Option is B

Step 1: Write the ellipse in standard form.
\[ 16x^2 + 9y^2 = 144 \Rightarrow \frac{x^2}{9} + \frac{y^2}{16} = 1 \]

Step 2: Identify major and minor axes.
Here, \[ a^2 = 16,\quad b^2 = 9 \] Since \( a^2>b^2 \) and is under \( y^2 \), the major axis is along the y-axis.

Step 3: Find \( c^2 \).
\[ c^2 = a^2 - b^2 = 16 - 9 = 7 \Rightarrow c = \sqrt{7} \]

Step 4: Write the foci.
\[ \text{Foci} = (0, \pm c) = (0, \pm \sqrt{7}) \]

Step 5: Conclusion.
\[ \boxed{(0, \pm \sqrt{7})} \]