Question

If the foot of the perpendicular from $(-4,5)$ to the straight line $3x-4y-18=0$ is $(\alpha ,\beta )$ then the value of $\alpha +\beta =$

• A. $1$
• B. $-1$
• C. $5$
• D. $-5$

Answer 1

The correct option is B $-1$

Slope of the line joining two points $(-4,5)$ and $(\alpha ,\beta )$ must be equal to $\frac{-4}{3}$.

$\Rightarrow \frac{\beta -5}{\alpha +4}=\frac{-4}{3}$

$4\alpha +3\beta =-1\dots \dots \left(1\right)$

And $(\alpha ,\beta )$ also satisfy the give equation of line.

$\Rightarrow 3\alpha -4\beta =18\dots \dots \left(2\right)$

From $1$ and $2$

$28(\alpha +\beta )-7\beta =-7$

$3(\alpha +\beta )-7\beta =18$

$\Rightarrow \alpha +\beta =-1$