Question

A line $y=mx+1$ is parallel to a line $3x+y+1=0$ when $m=a$ and perpendicular to the same line when $m=b$. Find the product of $a$ and $b$.

Answer 1

Two lines are parallel when their slopes are equal.

The slope of a line $y=mx+c$ is 'm'.

$3x+y+1=0\to y=-3x-1$

If $y=mx+c$ is parallel to $y=-3x-1$,
m = slope of $y=-3x-1$ = -3
So, a = -3

Two lines are perpendicular when the product of their slopes is -1.

So. $b\times -3=-1\to b=\frac{1}{3}$

$a\times b=-3\times \frac{1}{3}=-1$