Question

If the slope of one line of the pair of lines represented by $a{x}^2+10xy+y^2=0$ is four time the slope the other line, then ${}^{\prime }{a}^{\prime }$ equals to

• A. $1$
• B. $2$
• C. $4$
• D. $16$

Answer 1

The correct option is D $16$

Let the slopes of 2 lines which constitutes a pair of lines be $m_1$ and $m_2$.

For a pair of lines $a{x}^2+2hxy+b{y}^2=0$ ,

Sum of slopes of 2 lines, $m_1+m_2=\frac{-2h}{b}$ -(i)

Product of slopes of 2 lines, $m_1.m_2=\frac{a}{b}$ -(ii)

Using equations (i) and (ii) for pair of straight lines $a{x}^2+10xy+y^2=0$,

$m_1+m_2=\frac{-10}{1}$ $\Rightarrow$ $5m_1=-10$ (Since, one slope is 4 times the other)

$\Rightarrow$ $m_1=-2$ -(iii)

And, $m_1.m_2=\frac{a}{1}$ $\Rightarrow$ $4m_1^2=a$

$\Rightarrow$ $a=16$ (Using (iii))