34 days to CAT

Lesson 13 min· Section 1 of 4

The straight-up lesson, built for the exam.

Coordinate Geometry

Formula sheet

Everything for this chapter, in one place.

Points and lines


Distance between two points	$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Midpoint	$\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$
Slope	$m = \dfrac{y_2-y_1}{x_2-x_1}$
Line through a point	$y - y_1 = m(x-x_1)$
Intercept form	$\dfrac{x}{a} + \dfrac{y}{b} = 1$
Perpendicular lines	$m_1m_2 = -1$
Distance from a point to $ax+by+c=0$	$\dfrac{\lvert ax_0+by_0+c \rvert}{\sqrt{a^2+b^2}}$

Triangles on the plane


Area from vertices	$\dfrac{1}{2}\lvert x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2) \rvert$
Centroid	the average of the three vertices
Right-angled triangle	circumcentre is the midpoint of the hypotenuse, so $R = \dfrac{\text{hyp}}{2}$
Incircle of a right triangle	$r = \dfrac{a+b-c}{2}$

Quadrilaterals


Parallelogram $ABCD$	the diagonals bisect each other, so $A + C = B + D$
Hence the fourth vertex	$D = A + C - B$
Any line through the intersection of the diagonals	passes through the midpoint of each diagonal

Circles


Standard form	$(x-h)^2+(y-k)^2 = r^2$
General form	$x^2+y^2+Dx+Ey+F=0$ , centre $\left(-\frac{D}{2}, -\frac{E}{2}\right)$
Radius from the general form	$\sqrt{\left(\frac{D}{2}\right)^2+\left(\frac{E}{2}\right)^2 - F}$
Tangents from an outside point at angle $2\theta$	the point lies at distance $\dfrac{r}{\sin\theta}$ from the centre

Regions made of modulus


$\lvert x \rvert + \lvert y \rvert = c$	a square (tilted), diagonals $2c$ , area $2c^2$
Any modulus equation	split the plane into sign regions and draw a straight piece in each
An inequality	shade one side of each straight piece
Area	cut the shaded region into triangles and trapeziums; do not integrate

Why this chapter matters

0.4

questions in a typical Quant section

22nd

of 25 chapters

≈ 1 of 66 marks a section ride on it.

Asked in13 of 24 papers

20172025

Year by yearper section

2017: 12017: 1 per section
2018: 0.32018: 0.3 per section
2019: 0.72019: 0.7 per section
2020: 1.42020: 1.4 per section
2021: 02021: 0 per section
2022: 0.32022: 0.3 per section
2023: 0.72023: 0.7 per section
2024: 0.72024: 0.7 per section
2025: 0.32025: 0.3 per section

Question type17 asked

MCQ 71%Type-in 29%

Type-in answers carry no negative marking.

From every Coordinate Geometry question in CAT papers since 2017.