The gradient of one of the lines of $a x^2+2 h x y+b y^2=0$ is twice that of the other, then
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The gradient of one of the lines of $a x^2+2 h x y+b y^2=0$ is twice that of the other, then

(A) $h^2=a b$

(B) $h^2=a+b$

(C) $8 h^2=9 a b$

(D) $9 h^2=8 a b$

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Best answer

SOLUTION —

Here, $m_1+m_2=\frac{-2 h}{b}$ and $m_1 m_2=\frac{a}{b}$

Given that, $\quad m_1=2 m_2$

$\begin{array}{ll}\therefore & 3 m_2=\frac{-2 h}{b} \\\text { and } & 2 m_2^2=\frac{a}{b} \Rightarrow 2\left(\frac{-2 h}{3 b}\right)^2=\frac{a}{b} \\\Rightarrow & 8 h^2=9 a b\end{array}$

So, The correct option will be (C).

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