Q93 / 170 · Manipal (MET) 2020 · Mathematics
The locus of the extrimities of the latusrectum of the family of ellipses $b^2 x^2+y^2=a^2 b^2$ having a given major axis, is
- Correct answerA$x^2 \pm a y=a^2$
- B$y^2 \pm b x=a^2$
- C$x^2 \pm b y=a^2$
- D$y^2 \pm a x=b^2$
Solution
Given equation is
$b^2 x^2+y^2=a^2 b^2$
$\Rightarrow \quad \frac{x^2}{a^2}+\frac{y^2}{a^2 b^2}=1$ ....(i)
Above equation of an ellipse with semi-major axis (a) and semi-minor axis (ab).
N̦ow, ecentricity,
$e=1-\frac{a^2 b^2}{a^2} \Rightarrow b^2=1-e^2$ ....(ii)
Let $(x, y)$ be extrimities of latusrectum, then
$\begin{aligned} x & =a e \text { and } y= \pm \frac{a^2 b^2}{a} \\ \Rightarrow \quad \frac{x}{a} & =e \text { and } \frac{y}{a}= \pm b^2\end{aligned}$
From Eq. (ii), we get
$\pm \frac{y}{a}=1-\frac{x^2}{a^2} \Rightarrow \pm a y+x^2=a^2$
Hence, locus of latusrectum is $x^2 \pm a y=a^2$.
Hence, the correct answer is option: 'A'.