Question

If $(-4, 5)$ is the image of the point $(6, 1)$ with respect to the line $L$ , then $L$ is given by

• A. $5x + 2y = 1$
• B. $5x - 2y = 0$
• C. $5x - 2y + 1 = 0$
• D. $2x - 5y + 1 = 0$

Answer 1

Answer: (C)

$5x - 2y + 1 = 0$

Answer 2

We have, $P(-4, 5)$
is the image of $Q(6, 1)$ w.r.t. line $L$
So, mid-point of $PQ$ i.e., $R$ will lie on line $L$
$R=\left(\frac{-4+6}{2}, \frac{5+1}{2}\right) =\left(1, 3\right)$
Slope o f $PQ =\frac{5-1}{-4-6}=\frac{4}{-10}$
$=-\frac{2}{5}$
Since $L$ is perpendicular to $PQ$
$\therefore$ Slope of line $L=\frac{5}{2}$
Thus equation of line $L$ passing through $R\left( 1, 3\right)$ and having slope $\frac{5}{2}$ is given by
$y-3=\frac{5}{2}\left(x-1\right)$
$\Rightarrow 2y-6=5x-5$
$\Rightarrow 5x-2y+1=0$