Coordinate Geometry

Board: JEE Main | Exam: JEE Main

JEE Main study notes for Coordinate Geometry by Ajay Yadav.

Key Concepts

Straight Lines

Slope, angle between lines, distance formulas, foot of perpendicular, image of point. Family of lines.

Circles

General equation. Tangent, normal. Chord of contact. Length of tangent. Radical axis, radical center. Orthogonal circles. Family of circles.

Parabola

y²=4ax. Parametric: (at²,2at). Focal chord: t₁t₂=-1. Tangent: y=mx+a/m (slope form).

Ellipse

x²/a²+y²/b²=1. Eccentricity, foci, directrix. Parametric: (acosθ, bsinθ). Tangent: xx₁/a²+yy₁/b²=1. Condition of tangency: c²=a²m²+b².

Hyperbola

x²/a²-y²/b²=1. Asymptotes: y=±(b/a)x. Rectangular hyperbola: xy=c². Parametric: (ct, c/t).

Important Formulas

Circle center	(-g,-f) if eq: x²+y²+2gx+2fy+c=0
Parabola tangent	y = mx + a/m
Ellipse tangent	xx1/a²+yy1/b²=1
Hyperbola asymptote	y = ±(b/a)x

Solved Examples

Example 1: Find equation of circle with ends of diameter (1,2) and (3,4).

Solution: Center=(2,3), r=√1+1=√2. (x-2)²+(y-3)²=2.

Practice Questions

1. Find image of (1,2) in line 2x+3y=5.
2. Find tangents to circle x²+y²=25 with slope 2.
3. Find equation of parabola with focus (2,0) and directrix x=-2.
4. Find eccentricity of ellipse 5x²+9y²=45.
5. Find asymptotes of hyperbola x²/4-y²/9=1.

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