JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 6)
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 :
11x + 7y + 8 = 0 or 11x + 7y $$-$$ 15 = 0
11x $$-$$ 7y $$-$$ 8 = 0 or 11x + 7y + 15 = 0
2x $$-$$ 7y + 29 = 0 or 2x $$-$$ 7y $$-$$ 7 = 0
2x $$-$$ 7y $$-$$ 39 = 0 or 2x $$-$$ 7y $$-$$ 7 = 0
Explanation
_27th_July_Morning_Shift_en_6_2.png)
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 $$2x - 7y + 29 = 0$$
Comments (0)
