Analytical Geometry (Lines, Conics, Circles)
Syllabus Unit: Mathematics
Why it matters
Chapter overview
Analytical (coordinate) geometry translates algebra into pictures: straight lines and circles in two dimensions, the conic sections (parabola, ellipse, hyperbola) that share one unifying eccentricity-based definition, and an introduction to three-dimensional coordinate geometry — together forming a highly formula-based, high-yield block of the NDA Mathematics paper.
Mathematics & General Ability Mathematics 40% weightage
Live Animation1 / 4
Line: ax + by + c = 0
1Slope (m)(y2−y1)/(x2−x1), the line's steepness
2InterceptsWhere the line crosses the x-axis and y-axis
3Distance Formula√[(x2−x1)²+(y2−y1)²], distance between two points
4Angle Between Two Linestanθ = (m1−m2)/(1+m1m2)
1Coordinate Geometry & Straight Lines
Every straight line is fixed by just two numbers — its slope and one point — from which every other line fact follows.
Learning Objectives
- Compute the distance between two points and the slope of a line joining them.
- Write the equation of a line in slope-intercept, point-slope and two-point forms.
- Determine whether two lines are parallel (equal slopes) or perpendicular (product of slopes = −1).
- Find the angle between two lines and the distance of a point from a line.
Key Terms
Slope (m)(y2−y1)/(x2−x1); measures a line's steepness and direction.
Point-Slope Formy−y1 = m(x−x1), the equation of a line through (x1,y1) with slope m.
Perpendicularity ConditionTwo lines with slopes m1, m2 are perpendicular exactly when m1·m2 = −1.
Distance of a Point from a Line|ax1+by1+c| / √(a²+b²), for the line ax+by+c=0 and point (x1,y1).
Coordinate geometry represents every geometric shape as an algebraic equation, letting distance, angle and intersection problems be solved with formulas instead of construction.
Distance and slope
Distance = √[(x2−x1)²+(y2−y1)²] Slope m = (y2−y1)/(x2−x1)
Forms of a line's equation
- Slope-intercept: y = mx + c
- Point-slope: y − y1 = m(x − x1)
- Two-point: (y−y1)/(x−x1) = (y2−y1)/(x2−x1)
Parallel and perpendicular lines
Parallel: m1 = m2 Perpendicular: m1 · m2 = −1
Common MistakeA vertical line has undefined slope — remember it's still perpendicular to any horizontal line (slope 0), even though the m1·m2=−1 test can't be applied directly.
Angle between lines and distance from a point
tanθ = (m1−m2)/(1+m1m2) Distance of (x1,y1) from ax+by+c=0 = |ax1+by1+c|/√(a²+b²)