Tangent and Normal

Stand at a bend in a highway and look straight ahead along the road. The direction you are looking is the tangent direction — it is the direction the road is heading at that exact point. If you look sideways, perpendicular to the road, that is the normal direction — the direction the road is curving toward.

Now shrink the highway to a curve on paper and shrink yourself to a point on that curve. The tangent line at that point is the straight line that best approximates the curve right there — it has the same slope as the curve at that instant. The normal line is perpendicular to the tangent, pointing "into" or "out of" the curve's bend.

The derivative gives you the slope of the tangent. From there, the equation of the tangent line is just the point-slope form of a straight line — something you already know from coordinate geometry. The normal follows immediately, because perpendicular lines have slopes that are negative reciprocals.

Equation of the tangent

Take a curve y = f(x) and a point P = (a, f(a)) on it. The slope of the curve at P is f'(a). The tangent line passes through P with that slope.

Two special cases:

• Horizontal tangent: when f'(a) = 0, the tangent is the horizontal line y = f(a). This happens at maxima, minima, and points of inflection with zero slope.
• Vertical tangent: when f'(a) is undefined because the limit is \pm \infty, the tangent is the vertical line x = a. The curve y = x^{1/3} has a vertical tangent at the origin.

Finding the tangent to a specific curve

For the parabola y = x^2 at the point (3, 9):

The slope is f'(3) = 2(3) = 6. The tangent line is y - 9 = 6(x - 3), which simplifies to y = 6x - 9.

Check: at x = 3, the tangent gives y = 18 - 9 = 9, which matches the curve. And the slope of the tangent (6) matches the derivative (2 \times 3 = 6). The tangent line passes through the right point with the right slope — that is all a tangent needs to do.

Equation of the normal

The normal line at (a, f(a)) is perpendicular to the tangent. Two perpendicular lines have slopes that are negative reciprocals: if one has slope m, the other has slope -1/m.

For the parabola y = x^2 at (3, 9): the tangent has slope 6, so the normal has slope -1/6. The normal equation is y - 9 = -\frac{1}{6}(x - 3), or y = -\frac{x}{6} + \frac{19}{2}.

The tangent and normal are perpendicular at the point of contact and together they form a local coordinate system at that point — one axis along the curve, one across it.

Tangent and normal from an external point

Sometimes you need the tangent to a curve from a point that is not on the curve. For example: find the tangent to y = x^2 from the point (0, -1).

The idea: suppose the tangent touches the curve at some unknown point (t, t^2). The tangent line at that point is y - t^2 = 2t(x - t), which simplifies to y = 2tx - t^2. This line must pass through (0, -1):

So there are two tangent lines from (0, -1) to the parabola: y = 2x - 1 (touching at (1, 1)) and y = -2x - 1 (touching at (-1, 1)).

The general method: parametrise the contact point as (t, f(t)), write the tangent line equation, impose the condition that it passes through the external point, and solve for t.

Angle of intersection of two curves

When two curves cross, they form an angle at the crossing point. The angle of intersection is defined as the acute angle between their tangent lines at that point.

If curve y = f(x) has slope m_1 = f'(a) and curve y = g(x) has slope m_2 = g'(a) at the point of intersection (a, b), then the tangent of the angle \theta between the tangent lines is

This is the formula for the angle between two lines, applied to the two tangent lines. The absolute value ensures \theta is acute (between 0 and \pi/2).

Special cases:

• Orthogonal intersection: if m_1 m_2 = -1, the curves cross at right angles (\theta = 90°). Two curves that intersect at right angles everywhere are called orthogonal trajectories.
• Tangential intersection: if m_1 = m_2, the curves have the same tangent line at the crossing point (\theta = 0°). The curves touch but do not cross — or they cross so gently that their tangent lines coincide.

Finding where two curves meet

To use the angle formula, you first need the intersection point. Solve f(x) = g(x) to find x, then compute f'(x) and g'(x) at that point.

Take y = x^2 and y = x^3. Set x^2 = x^3: x^2(1 - x) = 0, so x = 0 or x = 1.

At x = 1: m_1 = 2(1) = 2 and m_2 = 3(1)^2 = 3. The angle is \tan\theta = \left|\frac{2 - 3}{1 + 6}\right| = \frac{1}{7}, so \theta = \tan^{-1}(1/7) \approx 8.13°.

At x = 0: m_1 = 0 and m_2 = 0. Both tangent lines are horizontal — the curves are tangent to each other at the origin. The angle is 0°.

Lengths: tangent, normal, subtangent, subnormal

At a point P = (a, f(a)) on a curve, the tangent and normal lines each hit the x-axis at some point. The distances between P, these intersection points, and the foot of the perpendicular from P to the x-axis define four geometric lengths.

Let m = f'(a) and k = f(a) (the y-coordinate of P). Drop a perpendicular from P to the x-axis; its foot is M = (a, 0).

Finding the four lengths

The tangent line at P = (a, k) is y - k = m(x - a). It meets the x-axis (y = 0) at the point T:

So T = \left(a - \frac{k}{m}, \, 0\right).

The normal line at P is y - k = -\frac{1}{m}(x - a). It meets the x-axis at N:

So N = (a + km, \, 0).

The four lengths:

The subtangent and subnormal are measured along the x-axis. The length of tangent and length of normal are measured along the tangent and normal lines themselves (from P to where they meet the axis).

Notice that the subnormal of the parabola y = ax^2 at any point is |y \cdot 2ax| = |ax^2 \cdot 2ax| = 2a^2|x|^3... but for the standard parabola y^2 = 4ax, the subnormal is constant. This is a famous property: for a parabola y^2 = 4ax, implicit differentiation gives 2y \cdot y' = 4a, so y' = 2a/y. The subnormal is |y \cdot y'| = |y \cdot 2a/y| = 2a — independent of the point. The subnormal of a parabola is always equal to the semi-latus rectum.

Computing the tangent and normal together

Common confusions

• "The tangent touches the curve at exactly one point." This is true for circles, but not in general. The tangent to y = x^3 at the origin is the line y = 0 (the x-axis), which also crosses the curve at the origin. A tangent is defined by its slope matching the curve's slope at the point — not by the number of intersection points.

• "The normal to y = f(x) has slope -f'(a)." No — it has slope -1/f'(a), the negative reciprocal, not the negative of the slope. The condition for perpendicularity is m_1 \cdot m_2 = -1, not m_1 + m_2 = 0.

• "Two curves always intersect at one angle." Only if they intersect at one point. If they intersect at multiple points (like y = x^2 and y = x), each intersection has its own angle — as you saw in Example 2, the angles were different at the two points.

• "The subtangent is the same as the length of tangent." They measure different things. The subtangent is the horizontal distance from T to M — it lies along the x-axis. The length of tangent is the actual distance from P to T along the tangent line — it is the hypotenuse of a right triangle. The subtangent is always shorter (or equal, if the tangent is horizontal).

• "If f'(a) = 0, the normal slope is -1/0, which is meaningless." When the tangent is horizontal, the normal is vertical: its equation is x = a. The formula -1/f'(a) formally gives \pm\infty, which corresponds to a vertical line. Similarly, if f'(a) = \pm\infty (vertical tangent), the normal is horizontal: y = f(a).

Going deeper

Why the subtangent matters

The subtangent |f(a)/f'(a)| has a clean geometric meaning: it tells you how far back you would need to start on the x-axis to "reach" the point P by walking up the tangent line. For exponential curves, the subtangent is constant. Consider y = e^x: f'(a) = e^a = f(a), so the subtangent is |f(a)/f'(a)| = 1 at every point. The tangent to e^x always crosses the x-axis exactly 1 unit to the left of the point of tangency. This is a defining property of the exponential function.

For the logarithmic curve y = \ln x: f'(a) = 1/a, so the subtangent is |f(a)/f'(a)| = |a \ln a|, which grows without bound. As a \to \infty, the subtangent grows faster than a — the tangent has to reach further and further back to hit the axis.

Orthogonal trajectories

Two families of curves are orthogonal trajectories of each other if every curve in one family intersects every curve in the other family at right angles. For example, the family of concentric circles x^2 + y^2 = r^2 and the family of straight lines y = mx through the origin are orthogonal trajectories — every radius of a circle is perpendicular to the circle at the point of intersection.

To find orthogonal trajectories of a family F(x, y, c) = 0:

1. Differentiate to get a relation between x, y, and dy/dx (eliminate c).
2. Replace dy/dx with -dx/dy (swap the slope with its negative reciprocal).
3. Solve the resulting differential equation.

The condition m_1 \cdot m_2 = -1 — the angle-of-intersection formula with \theta = 90° — is what drives the entire theory of orthogonal trajectories. It appears in physics (electric field lines and equipotential surfaces), fluid dynamics (streamlines and potential lines), and conformal mapping in complex analysis.

Tangent and normal in parametric form

If a curve is given parametrically as x = \phi(t), y = \psi(t), the slope at a point is \frac{dy}{dx} = \frac{\psi'(t)}{\phi'(t)}. The tangent at the point (\phi(t_0), \psi(t_0)) is

and the normal is

For a circle x = r\cos t, y = r\sin t: \frac{dy}{dx} = \frac{r\cos t}{-r\sin t} = -\cot t. The normal slope is \tan t. The normal line passes through (r\cos t, r\sin t) with slope \tan t — and if you extend it, it passes through the origin. The normal to a circle always points toward (or away from) the centre. That is a geometric fact you know already, but here it emerges from the calculus.

Where this leads next

• Derivative — the definition and meaning of f'(x), which is the slope used in every formula on this page.
• Straight Line — Forms — the six standard forms of a line equation, especially the point-slope form that generates the tangent and normal equations.
• Parabola — Tangent and Normal — tangent and normal lines applied specifically to parabolas, including the reflection property and the constant-subnormal result.
• Maxima and Minima — using the tangent's slope to find where a curve peaks and dips.
• Differentiability — when the tangent line exists and when it does not — corners, cusps, and vertical tangents.