Suppose $A, B$ are two points on $2 x-y+3=0$ and $P(1,2)$ is such that $P A=P B$. Then, the mid-point of $A B$ is

• Correct answer
  A

  $\left(\frac{-1}{5}, \frac{13}{5}\right)$
• B

  $\left(\frac{-7}{5}, \frac{9}{5}\right)$
• C

  $\left(\frac{7}{5}, \frac{-9}{5}\right)$
• D

  $\left(\frac{-7}{5}, \frac{-9}{5}\right)$

Solution

Therefore its slope is $\left(-\frac{1}{2}\right)$. $\therefore$ Equation of line $P M$ is $$ y-2=-\frac{1}{2}(x-1) $$
On solving Eqs. (i) and (ii), we get the mid-point of $A B$ is $M\left(-\frac{1}{5}, \frac{13}{5}\right)$.

Hence, the correct answer is option: 'A'.