Question

If the foot of perpendicular of the line drawn from $P(-3,5)$ on the line $x-y+2=0$ is $M(0,a)$ then value of $a$ is ___.

Answer 1

Slope of line $x-y+2=0$ is $1$, thus, slope of line perpendicular to $x-y+2=0$ is $-1$.

Since, the perpendicular line passes through $P(-3,5)$, its equation is as follows:

$y-y_1=m(x-x_1)\Rightarrow y-5=-1(x+3)\Rightarrow y-5=-x-3\Rightarrow x+y-5+3=0\Rightarrow x+y-2=0$

Now, foot of perpendicular will be the point of intersection of $x-y+2=0$ and $x+y-2=0$.

Adding both the equations, we get

$(x-y+2)+(x+y-2)=0\Rightarrow 2x=0\Rightarrow x=0$

Substituting the value of $x$ in $x-y+2=0$:

$0+y-2=0\Rightarrow y-2=0\Rightarrow y=2$

Therefore, $(x,y)=(0,2)$

Hence, coordinate of foot of perpendicular is $(0,2)$.

The value of $a$ is $2$.