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If the line $3 x+4 y+\lambda=0$ divides the distance between the lines $3 x+4 y+5=0$ and $3 x+4 y-5=0$ in the ratio of $3: 7$, then a value of $\lambda$ is
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Verified Answer
The correct answer is:
$2$
According to the given informations,
$\frac{\frac{|\lambda-5|}{\sqrt{3^2+4^2}}}{\frac{|\lambda+5|}{\sqrt{3^2+4^2}}}=\frac{3}{7} \Rightarrow \frac{|\lambda-5|}{|\lambda+5|}=\frac{3}{7}$
$\Rightarrow \quad \frac{5-\lambda}{5+\lambda}=\frac{3}{7} \quad\{\because \lambda \in(-5,5)\}$
$\begin{array}{ll}\Rightarrow & 35-7 \lambda=15+3 \lambda \\ \Rightarrow & 10 \lambda=20 \Rightarrow \lambda=2\end{array}$
Hence, option (b) is correct.
$\frac{\frac{|\lambda-5|}{\sqrt{3^2+4^2}}}{\frac{|\lambda+5|}{\sqrt{3^2+4^2}}}=\frac{3}{7} \Rightarrow \frac{|\lambda-5|}{|\lambda+5|}=\frac{3}{7}$
$\Rightarrow \quad \frac{5-\lambda}{5+\lambda}=\frac{3}{7} \quad\{\because \lambda \in(-5,5)\}$
$\begin{array}{ll}\Rightarrow & 35-7 \lambda=15+3 \lambda \\ \Rightarrow & 10 \lambda=20 \Rightarrow \lambda=2\end{array}$
Hence, option (b) is correct.
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