Question

Point $P(2,4)$ is translated through a distance $3\sqrt{2}$ units measured parallel to the line $y-x-1=0$, in the direction of decreasing ordinates, to reach at $Q$. If $R$ is the image of $Q$ with respect to the line $y-x-1=0$, then the coordinates of $R$ are

• A. $(-1,1)$
• B. $(5,7)$
• C. $(6,6)$
• D. $(0,0)$

Answer 1

The correct option is D $(0,0)$

Slope of the line $y-x-1=0$ is 1.
So, this line makes an angle of $\frac{\pi }{4}$ with x-axis.
The equation of a line through $P(2,4)$ and parallel to $y-x-1=0$ is
$\frac{x-2}{\cos \frac{\pi }{4}}=\frac{y-4}{\sin \frac{\pi }{4}}=-3\sqrt{2}$
The coordinates of $Q$ are given by
$\frac{x-2}{\cos \frac{\pi }{4}}=\frac{y-2}{\sin \frac{\pi }{4}}=-3\sqrt{2}\Rightarrow x=-1,y=1$
Thus, the coordinates of $Q$ are $(-1,1)$
Let R(h,k) is the image of $Q$ with respect to the line $y-x-1=0$.
$\frac{h+1}{-1}=\frac{k-1}{1}=-2\frac{(1+1-1)}{2}$
$\Rightarrow h=0,k=0$
Hence, the coordinates of $R$ are $(0,0)$.