Let $S(1,0)$ and $S^{\prime}(0,1)$ be the foci of an eflipse such that $\mathrm{SP}+\mathrm{S}^{\prime}…
Let $S(1,0)$ and $S^{\prime}(0,1)$ be the foci of an eflipse such that $\mathrm{SP}+\mathrm{S}^{\prime} \mathrm{P}=2$ for any point $\mathrm{P}$ on the ellipse. If $\mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $A^{\prime}\left(x_2, y_2\right)$ are the end points of the major axis of this ellipse, then $x_1+x_2=$
$-1 / 4$
$-1$
$1 / 4$
1
Solution
Since we have
$S P+S^{\prime} P=2 a=2$ (Given)
$\Rightarrow a=1$
And $S S^{\prime}=2 a e=\sqrt{1+1}=\sqrt{2} \Rightarrow e=\frac{1}{\sqrt{2}}$
$b^2=a^2\left(1-e^2\right) \Rightarrow b=\frac{1}{\sqrt{2}} < a$
$\therefore$ Length of semi major axis $=a=1$
Centre $=\left(\frac{1}{2}, \frac{1}{2}\right)$
End point of major axis $=\left(1+\frac{1}{2}, \frac{1}{2}\right)$ and $\left(-1+\frac{1}{2}, \frac{1}{2}\right)$
$\begin{aligned} & \text { i.e. }\left(\frac{3}{2}, \frac{1}{2}\right) \text { and }\left(-\frac{1}{2}, \frac{1}{2}\right) \\ & \therefore x_1+x_2=\frac{3}{2}-\frac{1}{2}=1 .\end{aligned}$