NDA 1 2025 Maths, question 54

Topic: Analytical geometry

Q54

Let $A(3, - 1)$ and $B(1,1)$ be the end points of line segment $AB$ . Let $P$ be the middle point of the line segment $AB$ . Let $Q$ be the point situated at a distance $\sqrt{2}$ units from $P$ on the perpendicular bisector line of $AB$ . What are the possible coordinates of $Q$ ?

(a) $(2,1)$
(b) $(3,1)$ Correct answer
(c) $(2,2)$
(d) $(1,3)$

Solution.

Answer: (b) $(3,1)$

The midpoint is $P = (2,0)$ .
Since the slope of $AB$ is $- 1$ , its perpendicular bisector has slope 1 and equation $y = x - 2$ .
A displacement $(t,t)$ from $P$ has length $\sqrt{2}|t|$ .
Hence $t = \pm 1$ , giving $Q = (3,1)$ or $(1, - 1)$ .
Only $(3,1)$ appears among the choices.