Question

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

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

Hint:

Complete the square in x to bring the conic to standard ellipse form.

Answer 1

The Correct Option is B

Step 1: Shift the origin
Put $X=x-3$. The conic becomes $25X^2+16y^2=400$.

Step 2: Foci in the shifted frame
This is $\dfrac{X^2}{16}+\dfrac{y^2}{25}=1$ with $c=\sqrt{25-16}=3$, so the foci are $(0,\pm3)$ in $(X,y)$.

Step 3: Shift back
Since $x=X+3$, the foci are $(3,\pm3)$, which is option (B).

Final Answer:
Shifting back from the centre gives foci at (3, plus or minus 3). \[ \boxed{\text{(B) }(3,\pm3)} \]