Question

If the image of point $P(2,3)$ in a line $L$ is $Q(4,5)$, then the image of point $R(0,0)$ in the same line is:

• A. $\left(2,2\right)$
• B. $\left(4,5\right)$
• C. $\left(3,4\right)$
• D. $\left(7,7\right)$

Answer 1

Answer: (D)

Mid-point of $P(2,3)$ and $Q(4,5)=(3,4)$
Slope of $PQ=1$
Slope of the line $L=-1$
Mid-point $(3,4)$ lies on the line $L$. Equation of line $L$,
$y-4=-1\left(x-3\right)\Rightarrow x+y-7=0$ ...(i)
Let image of point $R(0,0)$ be $S$ $\left(x_1,y_1\right)$
Mid-point of $RS=\left(\frac{x_1}{2}\frac{y_1}{2}\right)$
Mid-point $\left(\frac{x_1}{2},\frac{y_1}{2}\right)$ lies on the line (i)
$\therefore x_1+y_1=14$
Slope of RS =$\frac{y_1}{x_1}$
Since RS $\perp$ line L
$\therefore \frac{y_1}{x_1}\times \left(-1\right)=-1$
$\therefore x_1=y_1$
From (ii) and (iii),
$x_1=y_1=7$
Hence the image of $R=(7,7)$