Q97 / 160 · TS EAMCET 2019 (04 May Shift 2) · Mathematics
If $(1,-2)$ is the focus and $x+y-2=0$ is the directrix of the ellipse $17 x^2-2 x y+17 x^2-32 x+76 x+86-0$, then its eccentricity is
- A$\frac{1}{5}$
- B$\frac{1}{2}$
- Correct answerC$\frac{1}{3}$
- D$\frac{1}{4}$
Solution
Equation of ellipse whose focus is $(1,-2)$ and equation of directrix $x+y-2=0$ is
$$
\begin{aligned}
& (x-1)^2+(y+2)^2=\frac{e^2(x+y-2)^2}{2} \\
& 2\left(x^2-2 x+1+y^2+4 y+4\right) \\
= & e^2\left(x^2+y^2+4+2 x y-4 x-4 y\right) \\
\Rightarrow & x^2\left(2-e^2\right)-2 e^2 x y+\left(4 e^2-4\right) x \\
& +\left(2-e^2\right) y^2+\left(8+4 e^2\right) y+10-4 e^2=0
\end{aligned}
$$
Also given equation of ellipse
$$
17 x^2-2 x y+17 y^2-32 x+76 y+86=0
$$
Equation (i) and (ii) are identical
$$
\begin{aligned}
& \therefore \quad \frac{2-e^2}{17}=\frac{2 e^2}{2}=\frac{4+4 e^2}{32} \\
& \Rightarrow \quad 2-e^2=17 e^2 \quad \Rightarrow \quad 18 e^2=2 \\
& \Rightarrow \quad e^2=\frac{1}{9} \Rightarrow e=\frac{1}{3}
\end{aligned}
$$
Hence, the correct answer is option: 'C'.