Tangents and Normals

01 / UNDERSTAND & APPLY

1. A tangent describes local direction

A secant line joins two points of a curve. If one point approaches the other and the secant directions approach a unique limiting line, that line is a tangent. For a differentiable graph y = f(x), its finite slope at x = a is f′(a). The tangent passes through the actual curve point (a,f(a)); the derivative provides its direction, not its position.

The everyday phrase “a line that touches but does not cross” is unreliable. The tangent y = 0 to y = x³ at the origin crosses the curve there. A tangent can also meet the same curve again somewhere else. Tangency is a local statement about direction near a specified contact, not a global prohibition on intersections.

A normal is a line through the contact perpendicular to the tangent. When the tangent has a finite nonzero slope m, the normal slope is −1/m. If m = 0, however, the tangent is horizontal and the normal is vertical. Writing −1/0 is not a valid slope calculation. Use the geometric direction instead.

Before differentiating, verify that the proposed contact lies on the curve and in its domain. A line calculated at a forbidden x-value is not a tangent to the original graph even if a simplified expression has a removable extension there. The original domain determines which points actually exist.

02 / UNDERSTAND & APPLY

2. Tangent and normal formulas without exceptions hidden

For y = f(x) differentiable at a with m = f′(a), the tangent is y − f(a) = m(x − a). This formula remains valid at m = 0. A useful normal equation that also works when m = 0 is x − a + m(y − f(a)) = 0. When m is nonzero it rearranges to the familiar negative-reciprocal slope formula; when m is zero it becomes x = a.

The vector viewpoint explains both equations. The tangent direction is (1,m). A normal direction is (−m,1), whose dot product with (1,m) is zero. The normal line equation says that the displacement from the contact is perpendicular to the tangent direction. This avoids unnecessary division and makes horizontal cases routine.

These formulas assume the graph has a finite derivative. A vertical tangent, such as at the origin of y = x^(1/3), has no finite slope. Its tangent equation is x = 0 and its normal is y = 0. One can establish this with a limiting secant direction or by writing the same curve as x = y³ and differentiating with respect to y. An undefined derivative alone is not enough: a corner may have two different one-sided directions.

03 / UNDERSTAND & APPLY

3. A line through a point may touch elsewhere

When the problem gives a point through which a tangent passes, do not assume it is the contact. Introduce an unknown contact parameter a and write the tangent there. Substitute the given point into that line equation. The resulting equation determines possible contacts, which must then be checked in the original curve and line conditions.

This process can yield zero, one, or several tangents. A point on a curve may have additional tangents touching at other points. An external point may have more than two tangents to a general curve; the familiar two-tangent picture is specific to suitable conics and positions. Always solve the actual contact equation rather than relying on the circle diagram.

When the algebra squares an expression or clears a denominator, preserve restrictions and check candidate contacts afterward. Distinct contact parameters can occasionally produce the same tangent line, a bitangent. If the task asks for distinct lines, compare their equations instead of merely counting roots of the contact equation.

04 / UNDERSTAND & APPLY

4. Implicit curves and regular points

For a curve F(x,y) = 0 with differentiable F, implicit differentiation gives Fₓ + Fᵧ y′ = 0 wherever the curve can locally be treated as a differentiable graph y(x) and Fᵧ is nonzero. Thus y′ = −Fₓ/Fᵧ. A safer tangent equation at a regular point (a,b) is Fₓ(a,b)(x − a) + Fᵧ(a,b)(y − b) = 0.

The regular-point condition means the gradient (Fₓ,Fᵧ) is not the zero vector. This vector is perpendicular to the tangent, so it gives the normal direction directly. If Fᵧ = 0 but Fₓ ≠ 0, the tangent equation reduces to x = a and is vertical. The gradient approach handles that case without inventing an infinite numerical slope.

If both partial derivatives vanish, the displayed linear equation becomes 0 = 0 and gives no direction. Such a point requires a separate local analysis. The curve y² = x² has two crossing branches at the origin, with tangents y = x and y = −x. By contrast, y² = x³ has a cusp and a single tangent line y = 0. The zero gradient alone does not tell you which situation occurs.

The choice of defining equation matters at singular-looking points. Squaring a perfectly regular equation, as in F = (x² + y² − 1)², makes its gradient vanish all along the circle. That does not make the circle geometrically singular. Simplify the defining relation when possible and use a representation that retains the local direction information.

05 / UNDERSTAND & APPLY

5. Parametric curves: follow the velocity vector

For x = x(t), y = y(t), the vector (x′(t),y′(t)) points along the motion. At a regular parameter value t₀, meaning this vector is nonzero, it supplies a tangent direction even when the curve is not globally a graph. If x′(t₀) ≠ 0, the slope is dy/dx = y′(t₀)/x′(t₀). If x′ = 0 but y′ ≠ 0, the tangent is vertical.

A division-free tangent equation is x′(t₀)(y − y₀) − y′(t₀)(x − x₀) = 0. The normal equation is x′(t₀)(x − x₀) + y′(t₀)(y − y₀) = 0. These equations use the cross-product-zero and dot-product-zero conditions in the plane. They include horizontal and vertical tangents without separate reciprocal formulas.

If both parameter derivatives vanish, the parametrization is stationary, and the velocity formula gives no direction. It does not prove that no tangent exists. The parametrization x = t³, y = t⁶ has zero velocity at t = 0, but its curve is y = x² and has the horizontal tangent y = 0. Eliminate the parameter or examine the first nonzero terms.

One geometric point can correspond to several parameter values. At a self-intersection, those parameter values may provide different tangent directions. A complete answer should identify the requested branch or list each applicable tangent rather than silently choosing one value of t.

06 / UNDERSTAND & APPLY

6. Parallel, perpendicular and angle conditions

When a tangent must be parallel to a given nonvertical line of slope k, solve f′(a) = k and then compute the actual contact points and line equations. When a normal is parallel to that line, the tangent direction must be perpendicular to it. Use vectors if either line is vertical. Equality of slopes only determines direction, not coincidence.

For two nonvertical tangent slopes m₁ and m₂, the acute or zero angle φ between their unoriented lines satisfies tan φ = |(m₂ − m₁)/(1 + m₁m₂)| when the denominator is nonzero. If 1 + m₁m₂ = 0, the lines are perpendicular and φ = π/2. To avoid ambiguity, specify whether the problem requests the acute angle between lines or an oriented angle between moving directions.

The vector method works uniformly: for nonzero direction vectors u and v, cos φ = |u·v|/(|u||v|) gives the angle between unoriented lines in [0,π/2]. Without the absolute value it gives the angle between the chosen oriented vectors in [0,π]. At an intersection of two curves, first identify the correct common point, then evaluate each derivative at that point.

07 / UNDERSTAND & APPLY

7. Tangency, repeated intersections and local approximation

For a differentiable graph, subtract its tangent: R(x) = f(x) − f(a) − f′(a)(x − a). The remainder satisfies R(a) = 0 and R′(a) = 0. This expresses first-order agreement. If f is sufficiently smooth, higher derivatives determine whether the graph lies locally above, below or on opposite sides of the tangent. A positive second derivative at the contact places the curve locally above it; a cubic leading difference changes sign.

For a polynomial graph intersected with a line, tangency at a finite-slope contact implies that the difference polynomial has a repeated root. This gives an algebraic method: impose both P(a) = 0 and P′(a) = 0 for the intersection polynomial. A zero discriminant works directly when the resulting polynomial is quadratic, but do not apply a quadratic discriminant to a cubic or higher-degree equation.

The linear approximation f(a + h) ≈ f(a) + f′(a)h is a separate useful consequence. It becomes accurate relative to |h| as h tends to zero. The tangent is generally not an exact replacement for the whole curve, and the size of the error depends on curvature and distance from the contact. A tangent-line estimate far from the contact has no automatic accuracy guarantee.

A corner, a cusp and a vertical tangent are different phenomena. At a corner such as |x|, the one-sided slopes disagree. At a cusp such as y = |x|^(2/3), the slopes become unbounded in opposite signs but the limiting tangent line is vertical. At y = x^(1/3), both sides have positive unbounded slope and again a vertical tangent. Check line directions, not only the existence of a finite derivative.

08 / UNDERSTAND & APPLY

8. Original practice with contact checks

In each answer verify that the line contains the contact. When a vertical or horizontal line appears, write its equation directly instead of expressing it as an impossible slope.

09 / UNDERSTAND & APPLY

9. A compact checklist for every line

Write the contact, the direction and the line equation as three distinct steps. This prevents using a supplied external point as the contact or forgetting to evaluate the derivative at the correct parameter. Substitute the contact into your final equation; this catches many sign and intercept errors instantly.

Choose the representation that avoids artificial exceptions. Point-slope form is ideal for a finite derivative. The implicit gradient form handles vertical tangents. A parametric direction vector handles arbitrary regular parametrizations. At singular or stationary points, return to limits or the actual local branches rather than dividing zero by zero.

When asked for all tangents or normals satisfying a condition, solve for every allowed contact and deduplicate coincident lines. Check domain restrictions before accepting roots introduced by algebra. If the problem concerns angles, state the angle convention. If it concerns an approximation, state that the tangent is local and check how close the evaluation point is.

The normal passes through the same contact as the tangent. It is not generally the radius from the origin, except for special curves such as circles centered there. This distinction matters for ellipses, parabolas and shifted circles. Perpendicularity is a direction condition, not a license to choose an arbitrary line with the correct slope.