Tangents and Normals for JEE Main: Calculus Guide

Tangents and normals connect calculus (differentiation) to coordinate geometry (line equations), making them a natural bridge topic that JEE Main tests every session. The problems range from finding the equation of a tangent at a given point to locating where the tangent is horizontal, perpendicular to a given line, or where two curves share a common tangent. The calculation is always straightforward; the setup requires careful reading.

Slope of Tangent and Normal

The slope of the tangent to y = f(x) at x = a is f'(a). The normal at the same point is perpendicular to the tangent, so its slope is −1/f'(a) (provided f'(a) ≠ 0). If f'(a) = 0, the tangent is horizontal and the normal is vertical. If the curve is given parametrically as x = g(t), y = h(t), then dy/dx = (dy/dt)/(dx/dt) = h'(t)/g'(t). Implicit differentiation handles curves like x² + y² = r² (circle) or x² + xy + y² = 7, where solving for y is impractical. For the differentiation foundations see our continuity and differentiability guide.

Equations of Tangent and Normal

Tangent at (x₁, y₁): y − y₁ = f'(x₁)(x − x₁). Normal at (x₁, y₁): y − y₁ = −1/f'(x₁) × (x − x₁). Always verify that (x₁, y₁) lies on the curve before computing the slope. A common JEE question: find the equation of the tangent to y = x² at a point whose tangent passes through (0, −1). Set up: the tangent at (a, a²) is y − a² = 2a(x − a). Substitute (0, −1): −1 − a² = 2a(0 − a) = −2a². Solve: a² = 1, a = ±1. Two tangent lines result.

Horizontal and Vertical Tangents

Horizontal tangent: f'(x) = 0. Vertical tangent: f'(x) → ∞ or the derivative is undefined. For parametric curves, horizontal tangent when dy/dt = 0 (and dx/dt ≠ 0); vertical tangent when dx/dt = 0 (and dy/dt ≠ 0). At a cusp, both derivatives vanish simultaneously — the tangent is not defined in the usual sense. Finding all horizontal tangents means solving f'(x) = 0, which is a polynomial equation for polynomial functions.

Angle Between Two Curves

The angle between two curves at their intersection point is the angle between their tangents at that point. If m₁ and m₂ are the slopes of the tangents: tan φ = |m₁ − m₂|/(1 + m₁m₂). The curves are orthogonal (at right angles) when m₁m₂ = −1 — the product of their tangent slopes is −1. JEE regularly asks whether two conics intersect orthogonally, or to find the angle of intersection of two circles, a parabola and a line, etc.

Common and External Tangents

A common tangent to two curves touches both simultaneously. For two circles, there are external and internal common tangents — the number depends on whether the circles overlap. For a parabola and a line, finding where the line is tangent means imposing the discriminant = 0 condition on the combined equation. This discriminant method for tangency is used for circles, parabolas, ellipses, and hyperbolas, linking to our conic sections guide. After mastering tangent and normal problems, take a free mock test.