If the image of point $\mathrm{P}(2,3)$ in a line $\mathrm{L}$ is $\mathrm{Q}(4,5)$, then the image of point…
If the image of point $\mathrm{P}(2,3)$ in a line $\mathrm{L}$ is $\mathrm{Q}(4,5)$, then the image of point $\mathrm{R}(0,0)$ in the same line is:
$(2,2)$
$(4,5)$
$(3,4)$
$(7,7)$
Solution
Mid-point of $\mathrm{P}(2,3)$ and $\mathrm{Q}(4,5)=(3,4)$
Slope of PQ = 1
Slope of the line $\mathrm{L}=-1$
Mid-point $(3,4)$ lies on the line $L$.
Equation of line $\mathrm{L}$,
$
y-4=-1(x-3) \Rightarrow x+y-7=0
$
Let image of point $\mathrm{R}(0,0)$ be $\mathrm{S}\left(x_1, y_1\right)$
Mid-point of $\mathrm{RS}=\left(\frac{x_1}{2}, \frac{y_1}{2}\right)$
Mid-point $\left(\frac{x_1}{2}, \frac{y_1}{2}\right)$ lies on the line (i) $x_1+y_1=14$
Slope of RS $=\frac{y_1}{x_1}$
Since $\mathrm{RS} \perp$ line $\mathrm{L}$
$
\begin{aligned}
&\frac{y_1}{x_1} \times(-1)=-1 \\
&x_1=y_1
\end{aligned}
$
From (ii) and (iii),
$
x_1=y_1=7
$
Hence the image of $R=(7,7)$