Parabola — Tangent and Normal

A satellite dish is not shaped like a hemisphere. It is shaped like a paraboloid — a parabola rotated around its axis. Every incoming signal, arriving as a parallel wave, bounces off the dish and converges at a single point: the focus, where the receiver sits. The reason this works is a geometric property of the tangent line: at every point on the parabola, the tangent makes equal angles with the axis and with the line to the focus. That is the law of reflection, and proving it requires the tangent equation.

The tangent and normal at a point on a parabola appear in three forms — point form, parametric form, and slope form. Each form is useful in different situations: point form when you know the point of tangency, parametric form when you are working with the parameter t, and slope form when you know the slope of the tangent but not the point of contact. All three are derived from the same idea: the tangent is the line through a point on the curve with slope equal to the derivative at that point.

Equation of the tangent: point form

Take the parabola y^2 = 4ax and a point P(x_1, y_1) on it, so y_1^2 = 4ax_1.

To find the slope of the tangent, differentiate y^2 = 4ax implicitly with respect to x. Both sides are functions of x (with y depending on x), so:

Why: implicit differentiation treats y as a function of x and applies the chain rule to y^2, giving 2y \cdot dy/dx on the left side.

At the point P(x_1, y_1), the slope of the tangent is:

The equation of the tangent line through (x_1, y_1) with slope 2a/y_1 is:

Multiply both sides by y_1:

Since P is on the parabola, y_1^2 = 4ax_1. Substitute:

This formula has an elegant structure. In the equation y^2 = 4ax, replace one copy of y with y_1 (the y-coordinate of the point), and replace x with (x + x_1)/2 (the average of x and x_1). This pattern — called the T = 0 substitution — extends to all conic sections. For a circle x^2 + y^2 = r^2, the tangent at (x_1, y_1) is xx_1 + yy_1 = r^2, following exactly the same rule. Once you see the pattern, you can write down tangent equations for any conic almost by inspection.

Equation of the tangent: parametric form

Now express the tangent at the point t, where the point on the parabola is (at^2, 2at) (from the parametric form). Substitute x_1 = at^2 and y_1 = 2at into the point form yy_1 = 2a(x + x_1):

Divide both sides by 2a (positive, so no sign change):

This is one of the cleanest equations in conic geometry. It is linear in x and y, with the single parameter t controlling everything — the slope, the intercepts, and the position of the tangent line.

The slope of this tangent is found by rearranging: y = (1/t)x + at, so the slope is m = 1/t. This makes sense: at the vertex (t = 0), the tangent is vertical (undefined slope — the tangent is the line x = 0, which is the y-axis). As t grows, the tangent tilts to become more horizontal, with slope 1/t \to 0.

Checking the vertex tangent

At t = 0, the parametric tangent equation ty = x + at^2 becomes 0 = x + 0, i.e., x = 0. The tangent at the vertex is the y-axis — perpendicular to the axis of the parabola. This makes geometric sense: the vertex is the turning point of the curve, and at the turning point, the curve is perpendicular to the direction of its axis.

Equation of the tangent: slope form

Sometimes you know the slope of a desired tangent line but not the point where it touches the curve. For instance, you might need "the tangent to y^2 = 4ax with slope 3." The slope form handles this.

Start with a general line of slope m:

Substitute into y^2 = 4ax:

Why: bring all terms to one side to get a quadratic in x. The solutions of this quadratic give the x-coordinates of the intersection points of the line and the parabola.

This is a quadratic in x. The line is tangent to the parabola when it meets the curve at exactly one point — which happens when the discriminant of this quadratic is zero:

Expand the first square:

The 4m^2c^2 terms cancel:

Since a > 0 (the parabola is non-degenerate), this requires:

Deriving the point of tangency

With c = a/m, the quadratic m^2x^2 + (2mc - 4a)x + c^2 = 0 becomes:

Since the discriminant is zero, the single root is:

Then y = mx + a/m = m \cdot a/m^2 + a/m = a/m + a/m = 2a/m.

So the point of tangency is (a/m^2, 2a/m).

Connecting the three forms

The slope form and the parametric form are two views of the same thing. The parametric tangent at t has slope 1/t, so m = 1/t, i.e., t = 1/m. Substituting into the point of tangency (at^2, 2at):

This matches the slope form result exactly. The parameter t and the slope m are reciprocals: m = 1/t.

Equation of the normal

The normal at a point on a curve is the line perpendicular to the tangent at that point. If the tangent has slope m_T, the normal has slope m_N = -1/m_T (since perpendicular lines have slopes that multiply to -1).

Normal in parametric form

At the point t on y^2 = 4ax, the tangent slope is 1/t. The normal slope is therefore -t.

The normal passes through the point (at^2, 2at) with slope -t:

Normal in point form

At the point (x_1, y_1) on the parabola, the tangent slope is 2a/y_1, so the normal slope is -y_1/(2a). The normal equation is:

Multiplying through by 2a:

Normal in slope form

For a normal line y = Mx + c with slope M (using capital M to distinguish from the tangent slope m), the relationship M = -t gives t = -M. Substituting into the normal equation:

So the normal of slope M to y^2 = 4ax is:

This is a cubic in M when you set (x, y) to a specific external point — meaning from a given point, there can be up to three normals to the parabola. This is a striking difference from the circle, where through any external point there is exactly one normal (the line through the point and the centre). The three normals correspond to the three real roots of the cubic aM^3 + (2a - x)M + y = 0.

Properties of tangent and normal

The sub-tangent is bisected by the vertex

The sub-tangent is the projection on the axis of the segment of the tangent from the point of tangency to where it meets the axis. For the point P(at^2, 2at), the tangent ty = x + at^2 meets the x-axis (y = 0) at:

So the tangent meets the axis at the point T(-at^2, 0). The sub-tangent is the distance from T to the foot of the ordinate of P (the point (at^2, 0) directly below P):

The midpoint of this segment is at x = \frac{-at^2 + at^2}{2} = 0 — the vertex.

The vertex always bisects the sub-tangent. This property is useful for constructing tangent lines geometrically: to draw the tangent at a point P, drop a perpendicular from P to the axis, find the point T on the axis such that the vertex is the midpoint of T and the foot of the perpendicular, and then the line TP is the tangent.

The sub-normal is constant

The sub-normal is the projection on the axis of the segment of the normal from the point on the curve to where the normal meets the axis. For the normal y + tx = 2at + at^3 at the point t, set y = 0:

The foot of the normal on the axis is at (2a + at^2, 0). The sub-normal is:

The sub-normal is always 2a, regardless of which point on the parabola you choose. This constant equals the semi-latus rectum. It is a distinctive property of the parabola — for other conics, the sub-normal varies from point to point.

The reflection property

At any point P on the parabola, the tangent bisects the angle between the focal radius SP (the line from the focus to P) and the line from P parallel to the axis. This means:

The tangent at P makes equal angles with the axis and with the focal radius.

This is the reflection property: by the law of reflection (angle of incidence = angle of reflection), an incoming ray parallel to the axis reflects off the parabola through the focus. It is the mathematical reason behind every satellite dish, headlight reflector, and solar concentrator.

Here is the proof. Take the point P(at^2, 2at) on y^2 = 4ax. The tangent at P has slope 1/t, so it makes an angle \alpha with the positive x-axis where \tan\alpha = 1/t.

The focus is at S(a, 0). The slope of the focal radius SP is:

The angle \beta that SP makes with the positive x-axis satisfies \tan\beta = \frac{2t}{t^2 - 1}.

Now use the double-angle identity for tangent:

Why: in the double-angle formula, substitute \tan\alpha = 1/t, then simplify. The denominator 1 - 1/t^2 = (t^2 - 1)/t^2, and dividing 2/t by this gives 2t/(t^2 - 1).

So \tan 2\alpha = \tan\beta, which means \beta = 2\alpha (in the relevant range of angles). The focal radius makes angle 2\alpha with the axis, and the tangent makes angle \alpha. The tangent exactly bisects the angle between the focal radius and the axis direction.

Since the axis direction is also the direction of incoming parallel rays, the angle between the tangent and the incoming ray equals the angle between the tangent and the reflected ray (along SP). This is exactly the law of reflection.

Tangents at the endpoints of a focal chord are perpendicular and meet on the directrix

Take a focal chord with endpoints at parameters t_1 and t_2, where t_1 t_2 = -1 (the focal chord condition). The tangents at these two points are:

To find their intersection, subtract the first from the second:

Since t_1 \neq t_2, divide by (t_2 - t_1):

Substitute back into the first tangent equation:

Since t_1 t_2 = -1:

So the tangents meet at (-a,\; a(t_1 + t_2)) — a point whose x-coordinate is -a, which is the directrix.

The tangents at the endpoints of any focal chord meet on the directrix.

Furthermore, the slopes of the two tangents are 1/t_1 and 1/t_2. Their product:

Since the product of slopes is -1, the two tangents are perpendicular.

The tangents at the endpoints of a focal chord are always perpendicular to each other. This beautiful result connects the focal chord (a topic from position and parametric form) with the tangent line.

Worked examples

Common confusions

• "The tangent at (x_1, y_1) works for any point in the plane." The formula yy_1 = 2a(x + x_1) is derived using the condition y_1^2 = 4ax_1, which holds only when (x_1, y_1) is on the parabola. If you substitute a point that is not on the curve into the formula, you get the equation of the chord of contact — the line joining the two points where tangents from that external point touch the parabola. This is a different (and useful) object, covered in Parabola — Advanced.

• "A tangent touches the parabola at two coincident points." This is actually correct, not a confusion — but it often surprises students. The tangent is a line that meets the parabola in two points that have merged into one. Algebraically, the quadratic equation formed by substituting the tangent line into the parabola equation has a repeated root (discriminant zero). This is the precise meaning of "tangent": a line that touches but does not cross.

• "The normal always meets the parabola again." True for every point except the vertex. At the vertex (t = 0), the normal is the axis of the parabola (y = 0). Substituting y = 0 into y^2 = 4ax gives x = 0 — the vertex again. So the normal at the vertex only meets the parabola at one point.

• "There is exactly one tangent from any external point." From a point outside the parabola, you can draw two tangents (just as with a circle). From a point on the curve, there is one tangent. From a point inside the curve, there are zero tangent lines. The number of tangents from a point is determined by the position of the point with respect to the parabola.

• "The slope form y = mx + a/m is valid for all values of m." Not for m = 0. A horizontal tangent (m = 0) would require dy/dx = 2a/y = 0, which would need y \to \infty — meaning no horizontal tangent exists. The parabola y^2 = 4ax does not have a horizontal tangent line; its tangent slope ranges from +\infty (at the vertex) to 0 (asymptotically, far from the vertex), without ever reaching 0. The formula c = a/m correctly diverges as m \to 0, reflecting this geometric fact.

• "The normal at a point is always steeper than the tangent." Not necessarily. The tangent slope is 1/t and the normal slope is -t. At t = 1 (one endpoint of the latus rectum), the tangent has slope 1 and the normal has slope -1 — they have equal absolute slopes. For |t| < 1, the tangent is steeper; for |t| > 1, the normal is steeper.

Going deeper

Co-normal points

From an external point (h, k), the normals to y^2 = 4ax have feet at points whose parameters satisfy the cubic:

(This comes from substituting x = h and y = k into the normal y + tx = 2at + at^3 and rearranging as at^3 - ht + 2at - k = 0, which is at^3 + (2a - h)t - k = 0.)

This cubic has at most three real roots t_1, t_2, t_3. The three feet of the normals are called co-normal points. By Vieta's formulas applied to the cubic:

• t_1 + t_2 + t_3 = 0 (coefficient of t^2 is zero)
• t_1 t_2 + t_2 t_3 + t_3 t_1 = (2a - h)/a
• t_1 t_2 t_3 = k/a

The first relation, t_1 + t_2 + t_3 = 0, is the most important: the sum of the parameters of co-normal points is always zero. Since the y-coordinates are 2at_1, 2at_2, 2at_3, their sum is 2a(t_1 + t_2 + t_3) = 0. The sum of the ordinates of co-normal points is always zero — a result that appears frequently in competitive examinations.

The normal chord: second intersection formula

When a normal at the point t meets the parabola again at the point t', the relationship between t and t' is:

This follows from the co-normal relation. The normal at t meets the parabola at t and t', and the point (h, k) = (2a + at^2, -at^3 - 2at + at^3 + 2at) \ldots — more directly, you can derive it by solving the intersection equation.

From the normal at t: y = -tx + 2at + at^3. Substituting into y^2 = 4ax:

This yields a quadratic in x with roots at^2 and at'^2. By Vieta's formulas for this quadratic (or by direct computation):

This formula is worth memorising — it saves significant computation in problems that ask for the second intersection of a normal with the parabola.

The T = 0 pattern across conics

The tangent formula yy_1 = 2a(x + x_1) for the parabola is a special case of the general T = 0 rule for conics:

• Circle x^2 + y^2 = r^2: tangent at (x_1, y_1) is xx_1 + yy_1 = r^2.
• Parabola y^2 = 4ax: tangent at (x_1, y_1) is yy_1 = 2a(x + x_1).
• Ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1: tangent at (x_1, y_1) is \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1.
• Hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1: tangent at (x_1, y_1) is \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1.

The pattern is: replace x^2 by xx_1, y^2 by yy_1, 2x by (x + x_1), 2y by (y + y_1), and constants remain unchanged. This mnemonic, once internalised, lets you write tangent equations for any standard conic by inspection.

Where this leads next

You now have the full tangent and normal toolkit for the parabola. Here is where it connects:

• Parabola — Advanced — chord of contact, pole and polar, and applications of the tangent condition to intersection problems and locus questions.
• Parabola — Introduction — revisit the definitions and standard forms if needed.
• Parabola — Position and Parametric Form — the parametric coordinates and focal chord properties used throughout this article.
• Tangent and Normal to a Circle — the same tangent-normal ideas applied to the circle, with the T = 0 pattern in its simplest form.
• Derivative — the foundational concept that makes tangent lines rigorous: the slope of the tangent at a point is the derivative of the function at that point.