The equation of the pair of lines passing through the origin whose sum and product of slopes are…
The equation of the pair of lines passing through the origin whose sum and product of slopes are respectively the arithmetic mean and geometric mean of 4 and 9 is
$12 x^2-13 x y+2 y^2=0$
$12 x^2+13 x y+2 y^2=0$
$12 x^2-15 x y+2 y^2=0$
$12 x^2+15 x y-2 y^2=0$
Solution
Let $m_1$ and $m_2$ be the slopes of lines.
Then, $m_1+m_2=$ arithmetic mean $=\frac{13}{2}$ and $\quad m_1 m_2=$ geometric mean $=\sqrt{36}=6$.
Now, equation of the pair of lines passing through the origin is
$
\left(y-m_1 x\right)\left(y-m_2 x\right)=0
$
$
\Rightarrow y^2-\left(m_1+m_2\right) x y+m_1 m_2 x^2=0
$
Using Eq. (i) and (ii), we get
$
\begin{aligned}
y^2-\frac{13}{2} x y+6 x^2 & =0 \\
\Rightarrow \quad 12 x^2-13 x y+2 y^2 & =0
\end{aligned}
$