Q239 / 270 · NDA 2022 (Phase 1) · Mathematics

What is the equation of the locus of the mid-point of the line segment obtained by cutting the line $x+y=p$, (where $p$ is a real number) by the coordinate axes?

  • Correct answer
    A
    $x-y=0$
  • B
    $x+y=0$
  • C
    $x-y=p$
  • D
    $x+y=p$

Solution

Given equation of line is $\begin{aligned} & \mathrm{x}+\mathrm{y}=\mathrm{p} \\ & \therefore \frac{\mathrm{x}}{\mathrm{p}}+\frac{\mathrm{y}}{\mathrm{p}}=1\end{aligned}$ $\therefore$ line has equal intercept P with coordinate axes. $\therefore$ Midpoint of line segment $=\left(\frac{\mathrm{P}}{2}, \frac{\mathrm{P}}{2}\right)$ Let $(h, k)$ be the locus of the mid-point of the line segment $\begin{aligned} & \therefore \mathrm{h}=\frac{\mathrm{p}}{2} \text { and } \mathrm{k}=\frac{\mathrm{p}}{2} \\ & \therefore \mathrm{~h}=\mathrm{k}\end{aligned}$ So, equation of locus is $x=y$ i.e. $x-y=0$

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

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