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

Option: 1

\mathrm{2 x-7 y+29=0 \text { or } 2 x-7 y-7=0}


Option: 2

\mathrm{11 x-7 y-8=0 \text { or } 11 x+7 y+15=0}


Option: 3

\mathrm{11 x+7 y+8=0 \text { or } 11 x+7 y-15=0}


Option: 4

\mathrm{2 x-7 y-39=0 \text { or } 2 x-7 y-7=0}


Answers (1)

best_answer

Equation of reflected line PR is,

\mathrm{ \begin{aligned} & y-3=\frac{1-3}{-2-5}(x-5) \\ & \Rightarrow 7 y-21=2 x-10 \Rightarrow 2 x-7 y+11=0 \end{aligned} }

So, the equation of one directrix is

\mathrm{ 2 x-7 y+11=0 }                 ......(i)
Let the equation of other directrix be

\mathrm{ 2 x-7 y+\lambda=0 }                    ......(ii)
Now, distance between focus and directrix \mathrm{=\frac{8}{\sqrt{53}}}

\mathrm{ \begin{aligned} & \Rightarrow \frac{a}{e}-a e=\frac{8}{\sqrt{53}} \\ & \Rightarrow a\left(3-\frac{1}{3}\right)=\frac{8}{\sqrt{53}} \Rightarrow a=\frac{3}{\sqrt{53}} \end{aligned} }

\because \quad  Distance between two directrices \mathrm{=\frac{2 a}{e}}
\mathrm{ \begin{aligned} & \Rightarrow\left|\frac{11-\lambda}{\sqrt{2^2+7^2}}\right|=2 \times 3 \times \frac{3}{\sqrt{53}} \quad \text { (Using (i) and (ii)) } \\\\ \Rightarrow & |11-\lambda|=18 \\\\ \Rightarrow & 11-\lambda= \pm 18 \Rightarrow \lambda=11 \mp 18 \end{aligned} }

\mathrm{ \Rightarrow \lambda=-7 \text { or } \lambda=29 }

So, equation of other directrix can be,

\mathrm{ 2 x-7 y+29=0 \text { or } 2 x-7 y-7=0 }

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