Parabola — Position and Parametric Form

Take the parabola y^2 = 8x (so a = 2). The point (2, 4) lies on it — check: 16 = 8 \times 2. The point (1, 4) gives 16 for y^2 but only 8 for 4ax — the point has "too much y" for its x, so it sits outside the curve. The point (3, 2) gives 4 for y^2 but 24 for 4ax — the point has "too little y" for its x, so it sits inside the curve, between the two branches.

That observation — compare y_1^2 with 4ax_1 — gives a complete classification of every point in the plane with respect to the parabola. And once you see it, you start wanting a way to describe points that are guaranteed to be on the curve, without checking anything. That is exactly what parametric coordinates provide.

Position of a point with respect to a parabola

For the parabola y^2 = 4ax, define the expression

for a point P(x_1, y_1).

The logic is clean and geometric. The parabola itself is the set of all points where y^2 - 4ax = 0. At a point outside the parabola, y^2 is "too large" for the given x — the point is farther from the axis than the curve, so y_1^2 > 4ax_1 and S_1 > 0. At a point inside the parabola, y^2 is "too small" — the point sits between the curve and the axis, so y_1^2 < 4ax_1 and S_1 < 0.

Working through the signs carefully

Consider y^2 = 12x (so a = 3). Take three test points:

• P_1 = (3, 6): S_1 = 36 - 12(3) = 36 - 36 = 0. On the parabola.
• P_2 = (1, 5): S_1 = 25 - 12(1) = 25 - 12 = 13 > 0. Outside.
• P_3 = (4, 3): S_1 = 9 - 12(4) = 9 - 48 = -39 < 0. Inside.

Check P_1: 6^2 = 36 = 12 \times 3. Yes, (3, 6) satisfies the equation. Check P_2: at x = 1, the parabola has y = \pm\sqrt{12} \approx \pm 3.46. The point (1, 5) has y = 5 > 3.46 — it is indeed above (and therefore outside) the upper branch. Check P_3: at x = 4, the parabola has y = \pm\sqrt{48} \approx \pm 6.93. The point (4, 3) has y = 3 < 6.93 — it sits below the upper branch but above the axis, which is inside the parabola.

This position test extends to all four standard forms. For x^2 = 4ay (upward-opening), the relevant expression is S_1 = x_1^2 - 4ay_1, and the same sign rules apply. For y^2 = -4ax (left-opening), it is S_1 = y_1^2 + 4ax_1. The key is always: substitute the point into the equation rearranged as "\text{LHS} - \text{RHS} = 0" and check the sign.

Parametric coordinates

Here is the central idea of this article. A point (x, y) lies on the parabola y^2 = 4ax exactly when y^2 = 4ax. You want a way to generate such points using a single free parameter — one number that you can set to anything, and the resulting point is automatically on the curve.

The trick is to start with y. Set y = 2at for some real number t. The factor 2a is chosen for later convenience — it makes all the formulas clean. Then the condition y^2 = 4ax forces x:

Why: dividing both sides by 4a (which is positive) gives x = at^2 directly. The choice y = 2at was designed so that the 4a on the right cancels cleanly.

So for any real number t, the point (at^2, 2at) lies on the parabola y^2 = 4ax. Conversely, every point on the parabola corresponds to exactly one value of t: given a point (x_1, y_1) on the curve, t = y_1/(2a).

What t means geometrically

At t = 0, the point is (0, 0) — the vertex. As t increases from zero, the point moves along the upper branch:

For negative t, the point is on the lower branch (negative y). The magnitude |t| controls how far the point is from the vertex: larger |t| means farther along the curve. The sign of t tells you which branch.

Notice something satisfying: the endpoints of the latus rectum correspond to t = 1 and t = -1. At t = 1: (a, 2a). At t = -1: (a, -2a). These are exactly the latus rectum endpoints derived in the introduction.

Why parametric form is powerful

The power of parametric form is this: instead of carrying two coordinates (x, y) tied by the constraint y^2 = 4ax, you carry one free parameter t with no constraint at all. Two distinct points on the parabola are just two numbers t_1 and t_2. A chord joining them, a tangent at one of them, a normal — all reduce to expressions in t_1 and t_2, and the algebra simplifies dramatically.

For example, the slope of the chord joining the points t_1 and t_2:

Why: the numerator has 2a times the difference, the denominator has a times the difference of squares. The (t_2 - t_1) cancels, leaving a clean formula that depends only on the sum of the parameters.

This formula — slope of a chord = 2/(t_1 + t_2) — is so much cleaner than what you would get working with general coordinates (x_1, y_1) and (x_2, y_2). And this simplification compounds in every subsequent calculation.

Focal distance

The focal distance of a point on the parabola is its distance from the focus. This is one of the most frequently used quantities in parabola problems, and the formula is remarkably simple.

For the parabola y^2 = 4ax with focus S(a, 0), take a point P(x_1, y_1) on the curve. By the focus-directrix property, the distance from P to the focus equals the distance from P to the directrix x = -a:

(The absolute value is not needed because every point on the parabola has x_1 \geq 0 when a > 0, so x_1 + a > 0.)

This is beautiful in its simplicity. The distance from any point on the parabola to the focus is just a plus the x-coordinate of the point. No square roots, no Pythagorean theorem — the focus-directrix property does all the work.

In parametric form, the focal distance of the point t is:

Since 1 + t^2 \geq 1 for all real t, the minimum focal distance is a — achieved at t = 0, the vertex. The focus is always at least a units from the curve, and the vertex is the closest point.

Verifying with coordinates directly

You can also compute the focal distance using the distance formula directly, as a check. Take the point P(at^2, 2at) and the focus S(a, 0):

The expression under the square root is a perfect square — t^4 + 2t^2 + 1 = (t^2 + 1)^2 — which is why the formula comes out so clean. This is not a coincidence; it is the algebraic signature of the focus-directrix property.

Focal chord

A focal chord is any chord of the parabola that passes through the focus. These chords have remarkable algebraic properties that make them a favourite in examinations and in the theory of conics.

Let the endpoints of a focal chord be the points t_1 and t_2 on the parabola y^2 = 4ax, so the endpoints are P(at_1^2, 2at_1) and Q(at_2^2, 2at_2).

The fundamental condition: t_1 t_2 = -1

For the chord PQ to pass through the focus S(a, 0), the parameters t_1 and t_2 must satisfy a specific relationship. Here is the derivation.

The slope of the chord PQ is \frac{2}{t_1 + t_2} (derived above). The equation of the line through P(at_1^2, 2at_1) with this slope is:

For this line to pass through the focus S(a, 0), substitute x = a and y = 0:

Why: factor out a from the numerator on the right.

Factor (1 - t_1^2) = (1 - t_1)(1 + t_1):

Rewrite (1 - t_1) = -(t_1 - 1):

Cancel -2a from both sides (a > 0):

Cross-multiply:

This elegant condition means: if one endpoint has parameter t, the other has parameter -1/t. Since t and -1/t always have opposite signs, the two endpoints are always on opposite sides of the axis — one on the upper branch, one on the lower branch.

Length of a focal chord

Knowing t_1 t_2 = -1, you can find the length of a focal chord. The focal distances of the two endpoints are a(1 + t_1^2) and a(1 + t_2^2). Since the focus lies between the two endpoints on the chord, the chord length is the sum of the two focal distances:

Using t_2 = -1/t_1, we have t_2^2 = 1/t_1^2:

Recognise this as a perfect square:

Why: expand (t_1 + 1/t_1)^2 = t_1^2 + 2 \cdot t_1 \cdot (1/t_1) + 1/t_1^2 = t_1^2 + 2 + 1/t_1^2. The middle term is just 2.

So:

This formula gives the length of any focal chord in terms of the parameter of one endpoint.

The latus rectum is the shortest focal chord

The latus rectum is the focal chord perpendicular to the axis. It corresponds to the endpoints t_1 = 1 and t_2 = -1 (check: t_1 t_2 = -1), which give the points (a, 2a) and (a, -2a). Its length is:

This matches the length derived in the introduction. But there is a deeper fact: the latus rectum is the shortest of all focal chords.

Proof: from the formula \ell = a(t_1 + 1/t_1)^2, and by the AM-GM inequality:

Why: AM-GM states |u + v| \geq 2\sqrt{|uv|} when u and v have the same sign. Here t_1 and 1/t_1 always have the same sign (both positive or both negative), so the inequality applies.

Therefore \ell = a(t_1 + 1/t_1)^2 \geq a \cdot 4 = 4a, with equality when |t_1| = |1/t_1|, i.e., |t_1| = 1. That is exactly when the focal chord is the latus rectum.

Among all chords passing through the focus, the latus rectum is the shortest. Every other focal chord is longer.

Equation of the chord joining t_1 and t_2

For reference, the equation of the chord joining any two points t_1 and t_2 on the parabola y^2 = 4ax is:

This can be derived by writing the two-point form of the line using the points (at_1^2, 2at_1) and (at_2^2, 2at_2), and simplifying. For a focal chord (t_1 t_2 = -1), this becomes:

This is the general equation of any focal chord, expressed purely in terms of the sum t_1 + t_2.

Worked examples

Common confusions

• "The parameter t is always positive." No. The parameter t ranges over all real numbers. Positive t gives points on the upper branch (positive y), negative t gives points on the lower branch (negative y), and t = 0 gives the vertex. Restricting t to positive values would give you only half the parabola.

• "The parametric form only works for y^2 = 4ax." The idea extends to all four standard forms, with the appropriate modifications. For x^2 = 4ay (upward-opening), the parametric form is (2at, at^2) — the roles of x and y swap. For y^2 = -4ax (left-opening), you can use (-at^2, 2at), or equivalently, use the same (at^2, 2at) but with the understanding that a is replaced by -a in the equation.

• "S_1 > 0 means the point is inside the parabola." The opposite. S_1 > 0 means y_1^2 > 4ax_1, meaning the point has "too much y" for its x-value, placing it outside the curve. S_1 < 0 means the point is inside. This sign convention can be confusing because it is the reverse of the circle case (where a positive value of S_1 also means outside), but for different geometric reasons.

• "The focal distance formula SP = a + x_1 is always valid." The formula SP = a + x_1 is valid only for points on the parabola. It is a consequence of the focus-directrix property (SP = PM), which holds only when the point satisfies y^2 = 4ax. For a general point not on the curve, you must use the full distance formula SP = \sqrt{(x_1 - a)^2 + y_1^2}.

• "Any chord through the focus has t_1 t_2 = 1." The condition is t_1 t_2 = -1, not +1. The negative sign is essential — it ensures the two endpoints are on opposite sides of the axis (one t positive, the other negative). If t_1 t_2 = +1, both endpoints would be on the same branch, and the chord would not pass through the focus (which sits on the axis between the two branches).

Going deeper

The harmonic mean property of focal chords

For a focal chord with endpoints at parameters t and -1/t, the focal distances are l_1 = a(1 + t^2) and l_2 = a(1 + 1/t^2) = a(t^2 + 1)/t^2.

Compute the sum of reciprocals:

So the semi-latus rectum a satisfies:

This means the semi-latus rectum is the harmonic mean of the two focal segments (the two parts of the focal chord on either side of the focus). Since the harmonic mean of l_1 and l_2 is \frac{2l_1 l_2}{l_1 + l_2}, this gives \text{HM}(l_1, l_2) = 2a, which equals the semi-latus rectum (half the latus rectum length). This identity appears frequently in competitive problems and in the theory of optics.

Tangents at the endpoints of a focal chord

A result proved in Parabola — Tangent and Normal: the tangent lines at the two endpoints of a focal chord meet on the directrix. Furthermore, these two tangents are perpendicular to each other.

The proof is clean: the tangents at t_1 and t_2 intersect at the point (at_1 t_2, a(t_1 + t_2)). When t_1 t_2 = -1, the x-coordinate is a(-1) = -a — which is the directrix. And the product of the tangent slopes is (1/t_1)(1/t_2) = 1/(t_1 t_2) = -1, confirming perpendicularity.

Parametric form for shifted parabolas

If the vertex is at (h, k) instead of the origin, the parametric coordinates become:

Every formula from the standard case carries over with x replaced by (x - h) and y replaced by (y - k). The focal distance becomes a + (x_1 - h), the focal chord condition is still t_1 t_2 = -1, and all the algebra is the same — just shifted.

Connection to sum and product of roots

When a line y = mx + c intersects the parabola y^2 = 4ax, substituting gives a quadratic in x (or equivalently, a quadratic in t). The sum and product of the two values of t at the intersection points follow from Vieta's formulas. This link between the parametric description and the algebraic theory of quadratics is what makes the parametric form so useful — it converts geometry problems into algebra problems that Vieta's formulas can solve.

Where this leads next

You now have the position test, parametric coordinates, and the theory of focal chords. Here is where it connects:

• Parabola — Tangent and Normal — the equation of the tangent in slope form, point form, and parametric form, plus the condition for tangency and properties of the normal.
• Parabola — Advanced — chord of contact, pole and polar, and the director circle, all expressed cleanly in parametric coordinates.
• Parabola — Introduction — if you need to revisit the definition, standard forms, or terminology.
• Sum and Product of Roots — the algebraic machinery behind solving for intersection parameters t_1 and t_2 when a line meets a parabola.