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If the sum of the slopes of the lines given by $x^{2}-4 p x y+8 y^{2}=0$ is three times their product, then $p$ has the value
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Verified Answer
The correct answer is:
$\frac{3}{4}$
Given, equation of line is $x^{2}-4 p x y+8 y^{2}=0$
On comparing with $a x^{2}+2 h x y+b y^{2}=0$
$$
\Rightarrow \quad a=1, b=8,2 h=-4 p
$$
Given that, $m_{1}+m_{2}=3 m_{1} m_{2}$
$\Rightarrow \quad-\frac{2 h}{b}=\frac{3 a}{b}$
$\Rightarrow \quad-2 h=3 a$
$\Rightarrow \quad 4 p=3 a$
$\Rightarrow \quad p=\frac{3 a}{4}$
$\Rightarrow \quad p=\frac{3}{4} \quad[\because a=1]$
On comparing with $a x^{2}+2 h x y+b y^{2}=0$
$$
\Rightarrow \quad a=1, b=8,2 h=-4 p
$$
Given that, $m_{1}+m_{2}=3 m_{1} m_{2}$
$\Rightarrow \quad-\frac{2 h}{b}=\frac{3 a}{b}$
$\Rightarrow \quad-2 h=3 a$
$\Rightarrow \quad 4 p=3 a$
$\Rightarrow \quad p=\frac{3 a}{4}$
$\Rightarrow \quad p=\frac{3}{4} \quad[\because a=1]$
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