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Coordinate Geometry for CAT: Formulas, Weightage & Question Types

Regions built from modulus, shapes from vertices, circles and shortest distances.

Coordinate Geometry formula sheet

Everything CAT needs from Coordinate Geometry, in one place. Bookmark it for revision; the lesson in the app explains where each line comes from and when to reach for it.

Everything for this chapter, in one place.

Points and lines

Distance between two points(x2−x1)2+(y2−y1)2\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Midpoint(x1+x22, y1+y22)\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)
Slopem=y2−y1x2−x1m = \dfrac{y_2-y_1}{x_2-x_1}
Line through a pointy−y1=m(x−x1)y - y_1 = m(x-x_1)
Intercept formxa+yb=1\dfrac{x}{a} + \dfrac{y}{b} = 1
Perpendicular linesm1m2=−1m_1m_2 = -1
Distance from a point to ax+by+c=0ax+by+c=0∣ax0+by0+c∣a2+b2\dfrac{\lvert ax_0+by_0+c \rvert}{\sqrt{a^2+b^2}}

Triangles on the plane

Area from vertices12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\dfrac{1}{2}\lvert x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2) \rvert
Centroidthe average of the three vertices
Right-angled trianglecircumcentre is the midpoint of the hypotenuse, so R=hyp2R = \dfrac{\text{hyp}}{2}
Incircle of a right triangler=a+b−c2r = \dfrac{a+b-c}{2}

Quadrilaterals

Parallelogram ABCDABCDthe diagonals bisect each other, so A+C=B+DA + C = B + D
Hence the fourth vertexD=A+C−BD = A + C - B
Any line through the intersection of the diagonalspasses through the midpoint of each diagonal

Circles

Standard form(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2 = r^2
General formx2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0, centre (−D2,−E2)\left(-\frac{D}{2}, -\frac{E}{2}\right)
Radius from the general form(D2)2+(E2)2−F\sqrt{\left(\frac{D}{2}\right)^2+\left(\frac{E}{2}\right)^2 - F}
Tangents from an outside point at angle 2θ2\thetathe point lies at distance rsin⁡θ\dfrac{r}{\sin\theta} from the centre

Regions made of modulus

∣x∣+∣y∣=c\lvert x \rvert + \lvert y \rvert = ca square (tilted), diagonals 2c2c, area 2c22c^2
Any modulus equationsplit the plane into sign regions and draw a straight piece in each
An inequalityshade one side of each straight piece
Areacut the shaded region into triangles and trapeziums; do not integrate

The 5 Coordinate Geometry question types CAT repeats

Past CAT papers return to the same few shapes. Learn to recognise them and most questions become a known routine.

  1. 01

    The area of a region built from modulus

  2. 02

    A parallelogram or rectangle from its vertices

  3. 03

    A circle through given points

  4. 04

    A circle or locus hidden in an equation

  5. 05

    A shortest distance

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Coordinate Geometry in CAT: quick answers

How many Coordinate Geometry questions come in CAT Quant?
About 0.4 questions in a 22-question Quant section, worth roughly 1 marks. Coordinate Geometry appeared in 13 of the 24 CAT papers from 2017 to 2025, 17 questions in all, which ranks it 22 of 25 Quant chapters.
Are Coordinate Geometry questions in CAT MCQ or TITA?
29% of the Coordinate Geometry questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.
What types of Coordinate Geometry questions does CAT ask?
CAT keeps returning to 5 question shapes in Coordinate Geometry: The area of a region built from modulus; A parallelogram or rectangle from its vertices; A circle through given points; A circle or locus hidden in an equation; A shortest distance.

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