Question

Image of the point $(4,-3)$ with respect to the line $y=x,$ is

• A. $(-4,-3)$
• B. $(3,-4)$
• C. $(-3,4)$
• D. $(-3,-4)$

Answer 1

Answer: (C)

Let $P(x_1,y_1)$ is the image of the point $Q(4,-3)$ .
Midpoint of $PQ\left(\frac{x_1+4}{2},\frac{y_1-3}{2}\right)$ .
This pointlies on $y=x$ . $\therefore$ $\frac{x_1+4}{2}=\frac{y_1-3}{2}$
$\Rightarrow$ $x_1-y_1=-7$ ..(i)
Slope of $PQ=\frac{-3-y_1}{4-x_1}$ and slope of $y=x$
is 1 Since, PQ is perpendicular to
$y=x$
$\therefore$ $\left(\frac{-3-y_1}{4-x_1}\right).1=-1$
$\Rightarrow$ $-3-y_1=-4+x_1$
$\Rightarrow$ $x_1+y_1=1$ ...(ii)
On solving Eqs. (i) and (ii), $x_1=-3,y_1=4$
Hence, required point is $(-3,4)$ .