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. $(2,2)$
• B. $(4,5)$
• C. $(3,4)$
• D. $(7,7)$

Answer 1

The correct option is D $(7,7)$

Mid point of P(2,3) & Q(4,5)

= $\frac{2+4}{2},\frac{3+5}{2}=(3,4)$

Slope of PQ=1

slope of line L= -1

(3,4) lies on L

$\therefore$ equation of L.

$y-4=-1(x-2)$

$\Rightarrow x+y-7=0\cdots \left(1\right)$

Let image of R(0,0) be $S(x_1,y_1)$

mid point of $RS=(\frac{x_1}{2},\frac{y_1}{2})$ which lies on (1)

$x_1+y_1=14$

slope of $RS=\frac{y_1}{x_1}$

As $RS\perp$ line L

$\therefore \frac{y_1}{x_1}\times (-1)=-1$

$\therefore x_1=y_1$

So, $x_1=y_1=7$ Thus R S = (7,7)