A ray of light through (2,1) is reflected at a point P on the Y-axis and then passes throu…

Question

A ray of light through $(2,1)$ is reflected at a point $P$ on the $Y$-axis and then passes through the point $(5,3)$. If this reflected ray is the directrix of an ellipse with eccentricity $\dfrac13$ and the distance of the nearer focus from this directrix is $\dfrac8{\sqrt{53}}$, then the equation of the other directrix can be

Accepted Answer

The reflected ray (via the image of $(2,1)$ in the $Y$-axis, $(-2,1)$, collinear with $(5,3)$) has direction giving slope $\dfrac{3-1}{5+2}=\dfrac27$; the directrix line is $2x-7y+11=0$ (through $(5,3)$: $10-21+11=0$ ).

Distance from focus to this directrix: $\dfrac{a}{e}\cdot\text{(along normal)}=\dfrac{8}{\sqrt{53}}$ combined with $e=\frac13$ gives $a=\dfrac3{\sqrt{53}}$ and the directrix separation $\dfrac{2a}e=\dfrac{18}{\sqrt{53}}$.

Let the other directrix be $2x-7y+\lambda=0$; distance between the parallels:
$$\frac{|\lambda-11|}{\sqrt{53}}=\frac{18}{\sqrt{53}}\Rightarrow\lambda=29\text{ or }-7.$$

$$2x-7y+29=0\quad\text{or}\quad2x-7y-7=0.$$

Solutions · 1