Question

The foci of the conic \( 25x^2 + 16y^2 - 150x = 175 \) are

• A. (0, \( \pm \)3)
• B. (3, \( \pm \)3)
• C. (0, \( \pm \)5)
• D. (5, \( \pm \)5)

Hint:

For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b>a$, foci are $(0, \pm be)$.

Answer 1

The Correct Option is B

Step 1: Complete the square
$25(x^2 - 6x + 9) + 16y^2 = 175 + 225$
$25(x-3)^2 + 16y^2 = 400$
$\frac{(x-3)^2}{16} + \frac{y^2}{25} = 1$.
Step 2: Identify parameters
Ellipse with vertical major axis. $a^2 = 16, b^2 = 25$.
$e = \sqrt{1 - \frac{a^2}{b^2}} = \sqrt{1 - \frac{16}{25}} = \frac{3}{5}$.
Step 3: Find Foci
Foci relative to center $(3,0)$ are $(3, 0 \pm be)$.
$be = 25 \times \frac{3}{5} = 3$. (Wait, $b = \sqrt{25} = 5$, so $be = 5 \times \frac{3}{5} = 3$).
Step 4: Conclusion
Foci are (3, 3) and (3, -3).
Final Answer:(B)