Question

The point $P(3,3)$ is reflected across the line $y=-x$. Then it is translated horizontally $3$ units to the left and vertically $3$ units up. Finally it is reflected across the line $y=x$. What are the co-ordinates of the point after these transformations.

• A. $(0,-6)$
• B. $(0,0)$
• C. $(-6,6)$
• D. $(-6,0)$

Answer 1

The correct option is A $(0,-6)$

Let $Q(h,k)$ be the image of $P(3,3)$ w.r.t the line mirror $x+y=0$

$\frac{h-p}{a}=\frac{k-q}{b}=\frac{-2\times (ap+bq+c)}{a^2+b^2}$

$\frac{h-3}{1}=\frac{k-3}{1}=\frac{-2\left(6\right)}{2}$
$\Rightarrow h-3=k-3=-6$
$\Rightarrow h=-3,k=-3$
So, the image is $Q(-3,-3)$.
In the second transformation, point Q is translated horizontally 3 units to the left and vertically 3 units up.
So, the coordinates of new image is $(-3-3,-3+3)$ i.e. $(-6,0)$
In the third transformation, point $(-6,0)$ is reflected w.r.t the line $y=x$
So, the final image is $(0,-6)$.