CAT 2025 Slot 1 · Quant · Question 13
CAT 2025 Slot 1 previous year question on Coordinate Geometry (Basics), with the answer and a step-by-step solution.
The \left( x,y \right) coordinates of vertices P, Q and R of a parallelogram PQRS are \left( -3,-2 \right), \left( 1,-5 \right) and \left( 9,1 \right), respectively. If the diagonal SQ intersects the x-axis at \left( a,0 \right), then the value of a is
- A\frac{13}{4}
- B\frac{29}{9}
- C\frac{10}{3}
- D\frac{27}{7}
Pick an option, then check.
Answer & solution
Answer: Option B
In a parallelogram the diagonals bisect each other, so PR and QS share a midpoint.
Midpoint of PR =\left( \frac{-3+9}{2},\frac{-2+1}{2}\right)=\left( 3,-\frac{1}{2}\right)=A (Let).
The line through Q\left( 1,-5 \right) and A\left( 3,-\frac{1}{2}\right)has slope \frac{-\frac{1}{2}-\left( -5 \right)}{3-1}=\frac{9}{4}, so y+5=\frac{9}{4}\left( x-1 \right).
Setting y=0 because it passes through x axis:
5=\frac{9}{4}\left( x-1 \right),
so x-1=\frac{20}{9} and a=\frac{29}{9}.
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