The value of $λ$ for which the equation ${(10 x - 5)}^{2} + {(10 y - 7)}^{2} = λ^{2} {(5 x + 12 y + 7)}^{2}$
represents the parabola is

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$- \frac{10}{13}$

$\frac{10}{13}$

$\frac{1}{13}$

None of these

answer is A, B.

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Detailed Solution

Based on theory $\begin{aligned} (10 x - 5)^{2} + (10 y - 7)^{2} = λ^{2} (5 x + 12 y + 7)^{2} \\ \Rightarrow {(x - \frac{1}{2})}^{2} + {(y - \frac{7}{10})}^{2} = \frac{λ^{2}}{10^{2}} \frac{(5 x + 12 y + 7)^{2}}{(5)^{2} + (12)^{2}} \cdot [(5)^{2} + (12)^{2}] \\ \Rightarrow {(x - \frac{1}{2})}^{2} + (y - \frac{7}{10}) = \frac{λ^{2}}{(10)^{2}} (5^{2} + 12^{2}) \cdot \frac{(5 x + 12 y + 7)^{2}}{5^{2} + 12^{2}} \end{aligned}$

$will represent parabola of focus (\frac{1}{2}, \frac{7}{10}) and directrix 5 x + 12 y + 7 = 0 if$

$\begin{aligned} \Rightarrow \frac{1}{(10)^{2}} \cdot λ^{2} (5^{2} + 12^{2}) = 1 \\ \Rightarrow λ^{2} = \frac{(10)^{2}}{169} \\ \Rightarrow λ = {(\frac{10}{13})}^{2} \\ \Rightarrow λ = \pm \frac{10}{13} . \end{aligned}$