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

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

Hint:

For reflection problems on coordinate axes, reflect the point across the axis and join it to the final point to obtain the reflected ray. For ellipses, remember: \[ \text{Distance between directrices} = \frac{2a}{e}, \quad \text{Focus–directrix distance} = a\left(\frac{1}{e}-e\right). \]

Answer 1

The Correct Option is A

Step 1: Equation of the reflected ray (directrix)
The ray passes through $A(2,1)$, reflects at a point on the $y$-axis, and then passes through $B(5,3)$. Reflection at the $y$-axis is handled by reflecting point $A$ across the $y$-axis. \[ A'( -2,\,1 ) \] The reflected ray is the straight line passing through $A'$ and $B$. Slope: \[ m = \frac{3-1}{5-(-2)} = \frac{2}{7} \] Equation using point $B(5,3)$: \[ y - 3 = \frac{2}{7}(x - 5) \] \[ 7y - 21 = 2x - 10 \quad\Rightarrow\quad 2x - 7y + 11 = 0 \] Hence, the given directrix is: \[ D_1:\; 2x - 7y + 11 = 0 \] Step 2: Geometry of the ellipse
Eccentricity: \[ e = \frac{1}{3} \] For an ellipse: - Distance between the two directrices = $\dfrac{2a}{e}$ - Distance of nearer focus from a directrix: \[ d = a\left(\frac{1}{e} - e\right) \] Given: \[ d = \frac{8}{\sqrt{53}} \] Substitute $e = \frac{1}{3}$: \[ a\left(3 - \frac{1}{3}\right) = \frac{8}{\sqrt{53}} \] \[ a \cdot \frac{8}{3} = \frac{8}{\sqrt{53}} \quad\Rightarrow\quad a = \frac{3}{\sqrt{53}} \] Step 3: Distance between the two directrices
\[ \text{Distance} = \frac{2a}{e} = 2 \times \frac{3/\sqrt{53}}{1/3} = \frac{18}{\sqrt{53}} \] Step 4: Equation of the other directrix
Let the other directrix be: \[ D_2:\; 2x - 7y + c = 0 \] Distance between $D_1$ and $D_2$: \[ \frac{|11 - c|}{\sqrt{2^2 + (-7)^2}} = \frac{|11 - c|}{\sqrt{53}} \] Equating distances: \[ \frac{|11 - c|}{\sqrt{53}} = \frac{18}{\sqrt{53}} \] \[ |11 - c| = 18 \] \[ \Rightarrow\quad c = -7 \quad \text{or} \quad c = 29 \] Thus, the possible equations of the other directrix are: \[ 2x - 7y - 7 = 0 \quad \text{or} \quad 2x - 7y + 29 = 0 \] Among the given options, this corresponds to **Option (A)**. \[ \boxed{\text{Correct Answer: (A)}} \]