CAT 2025 Slot 1 QA Q13 Coordinate Geometry

Question 13 of the Quantitative Ability (QA) section, CAT 2025 Slot 1, with CAT's official answer and a worked solution.

The \((x, y)\) coordinates of vertices \(\mathrm{P}, \mathrm{Q}\) and R of a parallelogram PQRS are \((-3,-2),(1,-5)\) and \((9,1)\), respectively. If the diagonal SQ intersects the \(x\)-axis at \((a, 0)\), then the value of \(a\) is

A
\(\frac{13}{4}\)
B
\(\frac{29}{9}\)
C
\(\frac{10}{3}\)
D
\(\frac{27}{7}\)

Answer: (B) 29/9 Coordinate Geometry MCQ

Solution

In a parallelogram, \(S = P + R - Q = (-3 + 9 - 1, -2 + 1 + 5) = (5, 4)\).
Line \(SQ\) through \((1,-5)\) and \((5,4)\) has slope \(\frac94\).
At \(y = 0\): \(x = 1 + \frac{5 \times 4}{9} = \frac{29}{9}\).