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QJEEM21Ma6511: JEE Main 2021 Mathematics - Class 11 JEEM21Ma6511 Solution: A ray of light through (2, 1) is reflected at a point P on the…

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 13{1 \over 3} and the distance of the nearer focus from this directrix is 853{8 \over {\sqrt {53} }} , then the equation of the other directrix can be :

Multiple Choice Options

  • A 11x + 7y + 8 = 0 or 11x + 7y −- 15 = 0
  • B 11x −- 7y −- 8 = 0 or 11x + 7y + 15 = 0
  • C 2x −- 7y + 29 = 0 or 2x −- 7y −- 7 = 0
  • D 2x −- 7y −- 39 = 0 or 2x −- 7y −- 7 = 0

Step-by-Step Verified Solution

JEE Main 2021 (Online) 27th July Morning Shift Mathematics - Ellipse Question 48 English Explanation

Equation of reflected Ray

y−1=27(x+2)y - 1 = {2 \over 7}(x + 2)

7x−7=2x+47x - 7 = 2x + 4

2x−7y+11=02x - 7y + 11 = 0

Let the equation of other directrix is

2x−7y+λ2x - 7y + \lambda = 0

Distance of directrix from focus

ae−e=853{a \over e} - e = {8 \over {\sqrt {53} }}

3a−a3=8533a - {a \over 3} = {8 \over {\sqrt {53} }} or a=353a = {3 \over {\sqrt {53} }}

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

3a+a3=10a3=103×353=10533a + {a \over 3} = {{10a} \over 3} = {{10} \over 3} \times {3 \over {\sqrt {53} }} = {{10} \over {\sqrt {53} }}

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

=2×3×353=1853= 2 \times 3 \times {3 \over {\sqrt {53} }} = {{18} \over {\sqrt {53} }}

∣λ−1153∣=1853\left| {{{\lambda - 11} \over {\sqrt {53} }}} \right| = {{18} \over {\sqrt {53} }}

λ−11=18\lambda - 11 = 18 or −- 18

λ=29\lambda = 29 or −- 7

2x−7y−7=02x - 7y - 7 = 0 or 2x−7y+29=02x - 7y + 29 = 0

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