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If the equation $x^{2}-3 x y+\lambda y^{2}+3 x-5 y+2=0$ represents a pair of lines, where $\lambda$ is real number and $\theta$ is angle between them, then value of $\operatorname{cosec}^{2} \theta$ is (A) $10$ (B) $3$ (C) $9$ (D) $\frac{1}{3}$
If the equation $x^{2}-3 x y+\lambda y^{2}+3 x-5 y+2=0$ represents a pair of lines, where $\lambda$ is real number and $\theta$ is angle between them, then value of $\operatorname{cosec}^{2} \theta$ is (A) $10$ (B) $3$ (C) $9$ (D) $\frac{1}{3}$
Maths
If the equation $x^{2}-3 x y+\lambda y^{2}+3 x-5 y+2=0$ represents a pair of lines, where $\lambda$ is real number and $\theta$ is angle between them, then value of $\operatorname{cosec}^{2} \theta$ is (A) $10$ (B) $3$ (C) $9$ (D) $\frac{1}{3}$

The correct option is option(A) $10$

$x^{2}-3 x y+\lambda y^{2}+3 x-5 y+2=0$

Comparing with general equation $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$, we get $a=1, h=-3 / 2, b=\lambda, g=3 / 2, f=-5 / 2, c=2$

For pair of straight lines, $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$

$\begin{array}{l}
=>2 \lambda+45 / 4-25 / 4-9 N / 4-9 / 2=0 \\
\Rightarrow 2 \lambda+2 / 4-9 N / 4=0 \\
=>\lambda+2=0 \\
\Rightarrow \lambda=2
\end{array}$

angle between pair of straight lines is given by

$\begin{array}{l}
\tan \theta=2 \sqrt{\left(h^{2}-a b\right) /(a+b)} \\
{=2 \sqrt{(} 9 / 4-\lambda) /(1+\lambda)} \\
{=\sqrt{(9-4 \lambda) /(1+\lambda)}} \\
{=1 / 3} \\
{\cot \theta=3} \\
{\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta} \\
{=1+3^{2}} \\
{=} &{10}
\end{array}$

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