Question

If the straight lines $3x-5y=7$ and $4x+ay+9=0$ are perpendicular to one another, find the value of a.

Answer 1

Given straight lines are $3x-5y=7$ ... $\left(i\right)$ and

$4x+ay+9=0$ .... $\left(ii\right)$

Let $m_1$ and $m_2$ be the slopes of the two lines respectively.

Since they are perpendicular, we must have $m_1{m}_2=-1$

For the line $ax+by+c=0$, the slope is $m=-\frac{a}{b}$

So, the slopes of both the lines are:

$m_1=\frac{3}{5}$ and $m_2=-\frac{4}{a}$

$\Rightarrow \frac{3}{5}\times \frac{-4}{a}=-1$

$\Rightarrow a=\frac{12}{5}=2.4$

Thus, the value of $a$ is $2.4$.