If the slope of one line is twice the slope of other in the pair of straight lines $a x^2+2 h x y+b y^2=0$,…
If the slope of one line is twice the slope of other in the pair of straight lines $a x^2+2 h x y+b y^2=0$, then $8 h^2$ is equal to
- $-9 a b$
- $9 a b$
- $-7 a b$
- $7 a b$
Solution
Given pair of straight lines is
$a x^2+2 h x y+b y^2=0$
Here,
$m_1=2 m_2$ (given) ...(i)
Since,
$m_1+m_2=\frac{-2 h}{b}$ ...(ii)
and
$m_1 m_2=\frac{a}{b}$ ...(iii)
From Eqs. (i) and (ii)
$\begin{aligned} 2 m_2+m_2 & =-\frac{2 h}{b} \\ 3 m_2 & =\frac{-2 h}{b}\end{aligned}$ ...(iv)
From Eqs. (i) and (iii)
$\begin{aligned} 2 m_2 \cdot m_2 & =\frac{a}{b} \\ 2 m_2^2 & =\frac{a}{b}\end{aligned}$ $\ldots(v)$
From Eqs. (iv) and (v)
$\begin{gathered}2\left(\frac{-2 h}{3 b}\right)^2=\frac{a}{b} \\ 2 \cdot \frac{4 h^2}{9 b^2}=\frac{a}{b} \\ 8 h^2=9 a b\end{gathered}$
Asked in: AP EAMCET 2001
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