Question

If the coordinates of the foot of the perpendicular drawn from the point $(1,-2)$ on the line $y=2x+1$ is $(\alpha ,\beta )$, then the value of $|\alpha +\beta |$ is

Answer 1

Let $M$ be the foot of perpendicular drawn from the $P(1,-2)$ to $y=2x+1$
The equation of a line perpendicular to $y=2x+1$ is given by
$x+2y+\lambda =0...\left(1\right)$
This passes through $P(1,-2)$.

So,
$1-4+\lambda =0\Rightarrow \lambda =3$
On putting in equation $\left(1\right)$, we get $x+2y+3=0$
Point $M$ is the point of intersection of $2x-y+1=0$ and $x+2y+3=0$.
Solving these equations, we get
$x=-1,y=-1$
Therefore, the coordinates of the foot of perpendicular are $(-1,-1)$

Hence, the value of $|\alpha +\beta |=|-2|=2$