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 $\bot$ 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 $R = (7,7)$