Q78 / 155 · WBJEE 2016 · Mathematics
The equation of auxiliary circle of the ellipse $16 x^{2}+25 y^{2}+32 x-100 y=284$ is
- Correct answerA$x^{2}+y^{2}+2 x-4 y-20=0$
- B$x^{2}+y^{2}+2 x-4 y=0$
- C$(x+1)^{2}+(y-2)^{2}=400$
- D$(x+1)^{2}+(y-2)^{2}=225$
Solution
Given equation of ellipse is
$$
16 x^{2}+25 y^{2}+32 x-100 y=284
$$
Simplifying the given equation, we have ellipse as
$$
\frac{(x+1)^{2}}{25}+\frac{(y-2)^{2}}{16}=1
$$
So, the auxiliary equation of circle is
$$
\begin{array}{r}
(x+1)^{2}+(y-2)^{2}=25 \\
\Rightarrow \quad x^{2}+2 x+1+y^{2}-4 y+4=25
\end{array}
$$
$\therefore$
$x^{2}+y^{2}+2 x-4 y-20=0$
Hence, the correct answer is option: 'A'.