🔢 Quant

Coordinate Geometry

Distance, section formula, lines, and basic conics in the plane.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

4/5

of recent CAT years

% of Quant📈

~2%

of section questions

Est. hours⏱️

~6h

to master

2021

~1/22

2022

~1/22

2023

~1/22

2024

~1/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~0.3 per slot (2022: 0.3 · 2023: 0.3 · 2024: 0.7 · 2025: 0.3). Absent in 2021, then present every year since 2022 — usually as an area-of-a-region question (2024 ran one in two slots).

Coordinate Geometry

Algebra applied to the plane. Four formulas plus the slope idea handle almost every CAT question here.

The toolkit

Distance: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ .
Section formula: the point dividing the join in ratio $m:n$ is $\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$ ; midpoint is the $1:1$ case.
Slope: $m=\dfrac{y_2-y_1}{x_2-x_1}$ . Parallel ⇒ equal slopes; perpendicular ⇒ product of slopes $=-1$ .
Area from vertices: $\tfrac12\,\big|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\big|$ .

A worked example

For points $A(1,2)$ , $B(4,6)$ , and the triangle $O(0,0),P(4,0),Q(0,3)$ :

Distance $AB=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=\mathbf 5$ (a 3-4-5 in disguise).
Area of $\triangle OPQ$ — base $OP=4$ on the x-axis, height $OQ=3$ — is $\tfrac12\cdot4\cdot3=\mathbf 6$ .

🎯PYQ Evidence

On the coordinate plane, exploit the figure's defining property and complete the square to tame ugly inequalities. : a parallelogram's diagonals bisect each other, so the midpoint of PR equals the midpoint of QS — that pins down S = (5, 4), and the line SQ then meets the x-axis at a = 29/9. : completing the square turns x²+y²+4(x−y) into (x+2)²+(y−2)²−8, so −4 ≤ … ≤ 0 becomes a ring 4 ≤ (x+2)²+(y−2)² ≤ 8 centred at (−2, 2); the line through the centre halves it, giving area 2π. Identify what the shape is in coordinates, and the arithmetic becomes routine.

Common traps

Sign in the area formula. Take the absolute value; a negative just means the vertices were listed clockwise.
Vertical lines. Slope is undefined (not zero) when $x_1=x_2$ .
Section ratio order. $m:n$ weights the far point by $m$ — keep the order straight.

Checklist

Use distance and spot hidden 3-4-5 / 5-12-13
Perpendicular ⇒ slopes multiply to $-1$
Area via the vertex determinant, with $|\,\cdot\,|$
Mind undefined slope for vertical lines

Sample Questions

13 practice questions

Easy

What is the slope of the line represented by the equation x + 2y = 1?

Medium

The line 3x + 4y = 8 passes through all quadrants in the coordinate plane EXCEPT:

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 1

Hard

The (x, y) coordinates of vertices P, 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

CAT 2024 · Slot 1

Hard

In the XY-plane, the area, in sq. units, of the region defined by the inequalities y ≥ x + 4 and –4 ≤ $x^{2}$ + $y^{2}$ + 4(x – y) ≤ 0 is

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