Q86 / 160 · TS EAMCET 2013 · Mathematics
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
- Correct answerA$12 x^2-13 x y+2 y^2=0$
- B$12 x^2+13 x y+2 y^2=0$
- C$12 x^2-15 x y+2 y^2=0$
- D$12 x^2+15 x y-2 y^2=0$
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
$$
\begin{aligned}
y^2-\frac{13}{2} x y+6 x^2 & =0 \\
\Rightarrow \quad 12 x^2-13 x y+2 y^2 & =0
\end{aligned}
$$
Hence, the correct answer is option: 'A'.