Tangent and Normal — Advanced

Two parabolas, y = x^2 and y = (x - 3)^2, sit side by side on the plane. The first opens upward from the origin; the second opens upward from (3, 0). Somewhere between them, you can draw a single straight line that is tangent to both curves at once — touching each one without cutting through it.

Where is that line? What is its equation? And how do you find it systematically, instead of by guessing?

These are the questions this article answers. You already know how to write down the tangent to a single curve at a single point. The advanced problems are: tangents from a point that is not on the curve, tangents that are common to two curves, and the angle at which two curves meet when they cross. Each reduces to the same core move — writing down the tangent equation with an unknown parameter and then solving for the parameter — but the setups differ in instructive ways.

Tangent from an external point

When a point P lies on a curve, the tangent at P is unique (assuming the curve is smooth there). But when P lies off the curve, multiple tangent lines from P can touch the curve at different points.

Take the parabola y = x^2 and the external point P = (0, -1). The point P is below the vertex — it is not on the parabola. How many tangent lines to the parabola pass through P?

The method. Let the tangent touch the parabola at the point (a, a^2) for some unknown a. The slope of the parabola at x = a is \frac{dy}{dx} = 2a. So the tangent line at (a, a^2) is

which simplifies to

This tangent line must pass through P = (0, -1). Substitute x = 0, y = -1:

So a^2 = 1, giving a = 1 or a = -1. There are two tangent lines from (0, -1) to the parabola.

When a = 1: the tangent is y = 2x - 1, touching the parabola at (1, 1).

When a = -1: the tangent is y = -2x - 1, touching the parabola at (-1, 1).

The key insight is the unknown parameter a. You do not know where the tangent touches the curve, so you let that point be a variable. The tangent equation, expressed in terms of a, gives you a family of lines. Requiring the external point to lie on one of those lines produces an equation in a alone.

How many tangent lines? The number of tangent lines from an external point equals the number of real solutions of that equation. For a parabola, it is a quadratic in a, so at most two tangent lines. For higher-degree curves, you can get more.

Length of tangent, normal, subtangent, and subnormal

At a point P on a curve, the tangent and normal lines meet the x-axis at specific points. The distances involved have classical names that show up in competitive exams.

Let P = (x_0, y_0) lie on the curve y = f(x), with m = f'(x_0) being the slope of the tangent.

The tangent line at P has equation Y - y_0 = m(X - x_0). Setting Y = 0 gives the foot of the tangent on the x-axis:

The normal line at P has equation Y - y_0 = -\frac{1}{m}(X - x_0). Setting Y = 0 gives the foot of the normal on the x-axis:

Drop a perpendicular from P to the x-axis; its foot is M = (x_0, 0). Then:

These follow from elementary coordinate geometry. The subtangent and subnormal are just projections onto the x-axis; the "lengths" are the hypotenuses of the right triangles formed by P, M, and the respective foot.

A useful fact about the parabola. For y^2 = 4ax (the standard parabola), the subnormal is constant: it equals 2a, regardless of where you are on the curve. You can verify this directly. At the point (at^2, 2at), the slope is dy/dx = 1/t, and the subnormal is |y \cdot m| = |2at \cdot (1/t)| = 2a.

Angle of intersection of two curves

When two curves cross at a point, the angle of intersection is defined as the acute angle between their tangent lines at that point.

Suppose y = f(x) and y = g(x) meet at a point (x_0, y_0). The slopes of their tangents there are m_1 = f'(x_0) and m_2 = g'(x_0). The angle \theta between the two tangent lines satisfies

provided neither tangent is vertical and 1 + m_1 m_2 \neq 0. This is the standard formula for the angle between two lines with known slopes.

Special case: \theta = 90°. When the two tangent lines are perpendicular, \tan\theta is undefined and 1 + m_1 m_2 = 0, i.e., m_1 m_2 = -1. Curves that meet at right angles are called orthogonal at that point.

Special case: \theta = 0°. When the tangent lines are parallel (or identical), m_1 = m_2 and \tan\theta = 0. If the curves actually share the same tangent line, they are said to touch each other (or be tangent to each other).

Worked examples

Common tangents — the general strategy

The method in Example 2 generalises. To find common tangents to two curves C_1 and C_2:

1. Write the tangent to C_1 at a general point, parametrised by some variable a.
2. Write the tangent to C_2 at a general point, parametrised by b.
3. Set the slope and intercept of the first equal to the slope and intercept of the second.
4. Solve the resulting system in a and b.

Each solution gives one common tangent. The number of common tangents depends on the curves — two parabolas opening the same way might have one or two; a line and a circle can have zero, one, or two.

An alternate approach. Sometimes it is faster to write the tangent to one curve and then require it to be tangent to the second. If the tangent to C_1 at parameter a is y = mx + c (with m and c expressed in terms of a), substitute this line into the equation of C_2. The line is tangent to C_2 precisely when the resulting equation has a repeated root — i.e., its discriminant is zero. This gives an equation in a alone.

Condition for a line to be tangent

A line y = mx + c is tangent to a curve y = f(x) if and only if the equation f(x) = mx + c has a repeated root. Equivalently, the line touches the curve at exactly one point (counting multiplicity).

For the parabola y = x^2, the condition is: x^2 = mx + c must have a repeated root, so its discriminant m^2 + 4c = 0, giving c = -m^2/4. Every tangent to y = x^2 has the form y = mx - m^2/4.

For the circle x^2 + y^2 = r^2, substituting y = mx + c gives x^2(1+m^2) + 2mcx + c^2 - r^2 = 0. The discriminant-zero condition yields c^2 = r^2(1 + m^2), i.e., c = \pm r\sqrt{1+m^2}. This is the condition for a line to be tangent to a circle — a result you will use often.

Orthogonal curves

Two families of curves are orthogonal families if every curve of one family intersects every curve of the other at right angles. The classic example is the family of concentric circles x^2 + y^2 = r^2 and the family of straight lines through the origin y = kx. At every point of intersection, the radius and the line through the origin coincide, while the circle's tangent is perpendicular to the radius — so the curves meet at 90°.

To show that two families are orthogonal, the standard method is:

1. Find the differential equation of the first family by eliminating the parameter. For example, x^2 + y^2 = r^2 differentiates to 2x + 2y\,\frac{dy}{dx} = 0, so \frac{dy}{dx} = -\frac{x}{y}.

2. Find the differential equation of the second family. y = kx gives k = y/x, and differentiating gives \frac{dy}{dx} = k = \frac{y}{x}.

3. Check that the product of the slopes is -1: \left(-\frac{x}{y}\right)\left(\frac{y}{x}\right) = -1. Confirmed — the families are orthogonal.

Another important pair. The family of parabolas y^2 = 4ax (one for each a > 0) and the family of parabolas y^2 = 4b(2c - x) can form orthogonal systems under the right conditions. More common in exams is the family of rectangular hyperbolas xy = c^2 (for various c) versus the family x^2 - y^2 = a^2. Differentiating xy = c^2 gives dy/dx = -y/x; differentiating x^2 - y^2 = a^2 gives dy/dx = x/y. The product is (-y/x)(x/y) = -1, so the families are orthogonal.

Common confusions

• "A tangent from an external point always exists." Not necessarily. For a curve like y = e^x, every tangent line has positive slope, so a point directly above the curve with a required negative slope may have no tangent line through it. Whether tangents exist from an external point depends on the specific curve and point.

• "Common tangent means the tangent at a common point." A common tangent to two curves is a line tangent to both — it need not pass through a point where the curves meet. The two points of tangency are usually different points on different curves.

• "If two curves don't intersect, they can't have a common tangent." False. Two parabolas opening in opposite directions can have common tangent lines even though they never meet. The tangent is a line that touches each curve — the curves themselves do not need to touch each other.

• "Orthogonal curves means the curves don't intersect." The opposite — orthogonal curves do intersect, and they do so at right angles. "Orthogonal" refers to the angle at the intersection, not the absence of intersection.

• "The angle of intersection is the angle between the curves." It is the angle between the tangent lines at the point of intersection. The curves themselves are not straight, so you cannot define an angle between them directly. The tangent lines are the linear approximation, and the angle between those approximations is the angle of intersection.

Going deeper

Envelope of a family of lines

When you write the tangent to y = x^2 at a general point a as y = 2ax - a^2, you have described a one-parameter family of lines, parametrised by a. The parabola itself is the envelope of this family — the curve that every member of the family is tangent to. Envelopes appear across mathematics: the envelope of the normals to a curve is called the evolute, and it is the locus of centres of curvature. For the parabola y = x^2, the evolute turns out to be a cusp-shaped curve. Envelopes will appear again when you study families of circles and conics.

Orthogonal trajectories via differential equations

Finding orthogonal trajectories is one of the first real applications of differential equations. Given a family of curves with differential equation \frac{dy}{dx} = F(x, y), the orthogonal trajectories satisfy \frac{dy}{dx} = -\frac{1}{F(x, y)}. You replace the slope by its negative reciprocal, then solve the resulting ODE. This idea has physical meaning: in electrostatics, the electric field lines and the equipotential surfaces are orthogonal families. In fluid flow, the streamlines and the velocity potential curves are orthogonal. The mathematical structure you learned here — product of slopes equals -1 — is the same structure that organises these physical fields.

When implicit differentiation is essential

For curves given implicitly — say x^3 + y^3 = 3xy (the folium of Descartes) — you cannot write y = f(x) explicitly. But implicit differentiation still gives you \frac{dy}{dx} in terms of x and y. All the techniques above — angle of intersection, tangent from an external point, orthogonality — apply unchanged, with the slope expressed implicitly. The only additional care is that at a self-intersection of the curve, the implicit derivative gives both slopes simultaneously, and you must handle them separately.

Where this leads next

• Rate of Change — the derivative as a rate, applied to physics and geometry problems where quantities change together.
• Maxima and Minima — where the tangent line is horizontal — the critical points that optimisation problems revolve around.
• Mean Value Theorems — a deep guarantee that somewhere between two points, the tangent line is parallel to the secant. This is the theoretical backbone of calculus.
• Monotonicity — using the sign of the derivative to determine where a function is increasing or decreasing.
• Curvature — moving beyond the tangent line to measure how sharply a curve bends, using the second derivative.