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

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 $${1 \over 3}$$ and the distance of the nearer focus from this directrix is $${8 \over {\sqrt {53} }}$$, then the equation of the other directrix can be :

Accepted Answer

Equation of reflected Ray

$$y - 1 = {2 \over 7}(x + 2)$$

$7x - 7 = 2x + 4$

$2x - 7y + 11 = 0$

Let the equation of other directrix is

$$2x - 7y + \lambda $$ = 0

Distance of directrix from focus

$${a \over e} - e = {8 \over {\sqrt {53} }}$$

$$3a - {a \over 3} = {8 \over {\sqrt {53} }}$$ or $$a = {3 \over {\sqrt {53} }}$$

Distance from other focus $${a \over e} + ae$$

$$3a + {a \over 3} = {{10a} \over 3} = {{10} \over 3} \times {3 \over {\sqrt {53} }} = {{10} \over {\sqrt {53} }}$$

Distance between two directrix = $$ = {{2a} \over e}$$

$$ = 2 \times 3 \times {3 \over {\sqrt {53} }} = {{18} \over {\sqrt {53} }}$$

$$\left| {{{\lambda  - 11} \over {\sqrt {53} }}} \right| = {{18} \over {\sqrt {53} }}$$

$$\lambda  - 11 = 18$$ or - 18

$$\lambda  = 29$$ or -7

$2x - 7y - 7 = 0$ or $2x - 7y + 29 = 0$

Solutions · 1