Coordinate Geometry Cheat Sheet & Formula Guide - IIT-JEE MAINS 2027

Welcome to the ultimate LibreTexts-style chapter guide for Coordinate Geometry, specifically designed for the NTA IIT-JEE Mains 2027 examination pattern. This master reference guide compiles crucial formulas, canonical equations, standard forms, and high-yield key concepts derived from previous years' questions (PYQs).

Table of Contents

• 1. NTA Pattern & Chapter Weightage Analysis
• 2. Point & Line Fundamentals
• 3. Equations & Properties of Straight Lines
• 4. Pair of Straight Lines
• 5. Geometry of Circles & Tangents
• 6. General Conic Sections & Parabola
• 7. People Also Ask (FAQ)

1. NTA Pattern & Chapter Weightage Analysis

Coordinate Geometry constitutes approximately 20% to 25% of the Mathematics section in the IIT-JEE Mains exam. Questions frequently blend concepts across sub-topics, such as finding the tangent to a circle that is perpendicular to a given line, or evaluating conic section properties via locus problems. Mastering these standard forms and rapid calculation techniques is essential for securing high speed and accuracy under time constraints.

2. Point & Line Fundamentals

The foundation of 2D analytic geometry rests on basic distance metrics, section partitioning, and geometric centers of triangles.

Distance Formula

For two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), the Euclidean distance \(d\) is given by:

\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Section Formula

The coordinates of a point dividing the line segment joining \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) in the ratio \(m:n\) are:

• Internal Division: \(\bigl( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \bigr)\)
• External Division: \(\bigl( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \bigr)\)

Triangle Centers & Area

For a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

• Midpoint Formula: \(\bigl( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigr)\)
• Centroid Formula: \(\bigl( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \bigr)\)
• Area of Triangle (\(\Delta\)): \(\Delta = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\)

3. Equations & Properties of Straight Lines

Understanding linear relationships requires familiarity with various functional representations of line equations.

Slope & Angle Formulas

The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is defined as \(m = \tan\theta = \frac{y_2 - y_1}{x_2 - x_1}\).

The acute angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:

\[\tan\theta = | \frac{m_1 - m_2}{1 + m_1 m_2} |\]

• Parallel Condition: \(m_1 = m_2\)
• Perpendicular Condition: \(m_1 m_2 = -1\)

Standard Forms of Line Equations

• Slope-Intercept Form: \(y = mx + c\)
• Point-Slope Form: \(y - y_1 = m(x - x_1)\)
• Two-Point Form: \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\)
• Intercept Form: \(\frac{x}{a} + \frac{y}{b} = 1\)
• Normal Form: \(x \cos\alpha + y \sin\alpha = p\)
• General Form: \(Ax + By + C = 0\)

Perpendicular Distance

The perpendicular distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is expressed as:

\[d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\]

4. Pair of Straight Lines

Homogeneous Equation of Second Degree

The homogeneous equation \(ax^2 + 2hxy + by^2 = 0\) represents two straight lines passing through the origin.

• Angle between lines: \(\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}\)
• Coincident Lines Condition: \(h^2 = ab\)
• Perpendicular Lines Condition: \(a + b = 0\)

General Second Degree Equation

The general second-degree equation \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) represents a pair of straight lines if and only if the determinant condition holds:

\[\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0\quad \text{and}\quad h^2 \ge ab\]

5. Geometry of Circles & Tangents

Standard and General Equations

• Standard Form: \((x - h)^2 + (y - k)^2 = r^2\) with center \((h, k)\) and radius \(r\).
• General Form: \(x^2 + y^2 + 2gx + 2fy + c = 0\) with center \((-g, -f)\) and radius \(r = \sqrt{g^2 + f^2 - c}\).
• Diameter Form: \((x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0\) for endpoints \((x_1, y_1)\) and \((x_2, y_2)\).

Tangent Conditions

• At Point \((x_1, y_1)\) on Circle \(x^2 + y^2 = r^2\): \(xx_1 + yy_1 = r^2\)
• Slope Form Condition: For line \(y = mx + c\) to be tangent to \(x^2 + y^2 = r^2\), \(c = \pm r\sqrt{1 + m^2}\).

6. General Conic Sections & Parabola

General Conic Identification

A conic section is defined by its eccentricity \(e\):

• Parabola: \(e = 1\)
• Ellipse: \(e < 1\)
• Hyperbola: \(e > 1\)

For the second-degree equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), classify using the discriminant \(\Delta = B^2 - 4AC\):

• \(\Delta < 0\): Ellipse (or circle/point/ellipse degenerate)
• \(\Delta = 0\): Parabola (or parallel lines)
• \(\Delta > 0\): Hyperbola (or intersecting lines)

7. People Also Ask (FAQ)

What is the weightage of Coordinate Geometry in IIT-JEE Mains 2027?

Coordinate Geometry typically carries around 20-25% weightage in the Mathematics section of JEE Mains, accounting for 5 to 7 direct or multi-concept questions.

How do I identify if a general second-degree equation represents a pair of straight lines?

Check if \(\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0\) and ensure \(h^2 \ge ab\). If \(\Delta eq 0\), it represents a general conic section.

What is the slope condition for tangency to a circle?

For a circle \(x^2 + y^2 = r^2\), the straight line \(y = mx + c\) is tangent if \(c = \pm r\sqrt{1 + m^2}\).