NDA Maths · Conics

Parabola — Equation, Properties & Latus Rectum

A parabola (e = 1) is the set of points equidistant from a focus and a directrix; its standard form y² = 4ax fixes the vertex, focus, directrix, axis, and latus rectum from the single number a.

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Why this matters

13 PYQs. Once you match the equation to one of the four standard orientations and read off a, every property — focus, directrix, latus rectum (4a), focal distance — is immediate. Sign and orientation are where the marks are won or lost.

Concept 1 of 4

Standard Forms & Their Elements

Intuition

There are four standard parabolas through the origin, one opening each way. Spotting which variable is squared (and the sign) tells you the axis and the direction it opens; the coefficient gives a, and a fixes the focus and directrix.

Definition

With vertex at the origin and $a > 0$ :

$y^2 = 4ax$ : opens right, focus $(a,0)$ , directrix $x=-a$ , axis $y=0$ .
$y^2 = -4ax$ : opens left, focus $(-a,0)$ , directrix $x=a$ .
$x^2 = 4ay$ : opens up, focus $(0,a)$ , directrix $y=-a$ .
$x^2 = -4ay$ : opens down, focus $(0,-a)$ , directrix $y=a$ .

The squared variable names the axis; the sign of the linear term gives the direction. From a focus and directrix, the vertex is their midpoint.

Standard parabola

y^2 = 4ax: \ \text{focus } (a,0), \ \text{directrix } x = -a

Worked example

Find the focus and directrix of

x^2 = -3y

From the bank · past-year question

Example 1ConicsMODERATE

What is the equation of the parabola with focus

(-3,0)

and directrix

x-3=0

[Q84 · Apr · 2022]

The sign of the linear term sets the direction

x^2 = -3y

opens DOWNWARD (negative coefficient), so the focus is below the vertex and the directrix above. Reading it as upward flips both — the most common parabola error.

Directrix of $y^2=4ax$ is $x=-a$ , on the OTHER side of the vertex

The focus is at

(a,0)

and the directrix is

x = -a

— the directrix sits on the opposite side of the vertex from the focus. Writing the directrix as

x = a

(the focus's coordinate) is a frequent slip.

Drill 3 more on standard forms & their elements

Concept 2 of 4

The Latus Rectum

Intuition

The latus rectum is the chord through the focus perpendicular to the axis — the parabola's 'width' at the focus. Its length is exactly 4a, the same coefficient that appears in the standard form, and its endpoints determine the parabola.

Definition

For $y^2 = 4ax$ , the latus rectum is the vertical chord through the focus $(a,0)$ , with length $4a$ and endpoints $(a, \pm 2a)$ . Knowing the latus rectum's endpoints fixes the focus (their midpoint) and the value of $a$ ; two parabolas can share the same latus rectum (opening in opposite directions).

Length of latus rectum

\text{LR} = 4a, \quad \text{endpoints } (a, \pm 2a)

Worked example

Find the length of the latus rectum of

y^2 = 12x

From the bank · past-year question

Example 2ConicsEASY

A parabola passes through

(1,2)

and satisfies

\frac{dy}{dx}=\frac{2y}{x},\ x>0,y>0

What is the length of latus rectum of the parabola?

[Q52 · Sep · 2023]

Parabola latus rectum is $4a$ , not $2a$

The full chord through the focus is

4a

long — its half-length (focus to one endpoint) is

2a

. Quoting

2a

as the latus rectum halves the answer. Read

4a

straight off the coefficient: for

y^2 = 12x

4a = 12

, so the latus rectum is

12

Drill 4 more on the latus rectum

Concept 3 of 4

Focal Distance & Focal Chords

Intuition

The defining property says a point's distance to the focus equals its distance to the directrix — so the focal distance is just the point's coordinate plus a. This turns 'distance to focus' questions into one addition.

Definition

For a point $(x_1, y_1)$ on $y^2 = 4ax$ :

Focal distance $= x_1 + a$ (the distance to the directrix $x=-a$ , by the focus–directrix property).
For $x^2 = 4ay$ , the focal distance is $y_1 + a$ .
A focal chord passes through the focus; the latus rectum is the shortest focal chord.

Focal distance

\text{focal distance of } (x_1,y_1) \text{ on } y^2=4ax = x_1 + a

Worked example

A point on

y^2 = 8x

has focal distance 6. Find its x-coordinate.

From the bank · past-year question

Example 3ConicsMODERATE

In the parabola

y^2=8x

, the focal distance of a point

P

lying on it is 8 units. Which of the following statements is/are correct? (A) The coordinates of

P

can be

(6,4\sqrt{3})

. (B) The perpendicular distance of

P

from the directrix of the parabola is 8 units. Select the correct answer using the code given below:

[Q64 · Apr · 2024]

Focal distance is $x_1 + a$ , not $x_1 - a$

By the focus–directrix property, the distance to the focus equals the distance to the directrix

x=-a

, which is

x_1 + a

(you ADD

a

). Using

x_1 - a

(distance to the focus's x-coordinate) is wrong because the directrix, not the focus, sets the measurement.

Concept 4 of 4

Tangents & Chords of a Parabola

Intuition

Tangent and chord questions reduce to substituting a line into the parabola and using the slope or the angle. A tangent of slope m to y² = 4ax has a fixed form; chords through the vertex or focus give clean intersections.

Definition

**Tangent of slope $m$ ** to $y^2 = 4ax$ : $y = mx + \dfrac{a}{m}$ (touch point $\left(\tfrac{a}{m^2}, \tfrac{2a}{m}\right)$ ). A tangent inclined at angle $\theta$ has $m=\tan\theta$ .
Chord through the vertex at angle $\theta$ : substitute $y = x\tan\theta$ to find where it meets the curve.
Two parabolas $y^2=4ax$ and $x^2=4ay$ meet at $(0,0)$ and $(4a,4a)$ , both on the line $y=x$ .

Tangent of slope m

y = mx + \dfrac{a}{m} \quad (\text{to } y^2 = 4ax)

Worked example

Find the equation of the tangent of slope

2

y^2 = 8x

From the bank · past-year question

Example 4ConicsEASY

A tangent to the parabola

y^2 = 4x

is inclined at an angle

45°

with the positive direction of

x

-axis. What is the point of contact of the tangent and the parabola?

[Q58 · Apr · 2025]

Drill 2 more on tangents & chords of a parabola

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Standard Forms & Their Elements
Standard parabola
$y^2 = 4ax: \ \text{focus } (a,0), \ \text{directrix } x = -a$
The Latus Rectum
Length of latus rectum
$\text{LR} = 4a, \quad \text{endpoints } (a, \pm 2a)$
Focal Distance & Focal Chords
Focal distance
$\text{focal distance of } (x_1,y_1) \text{ on } y^2=4ax = x_1 + a$
Tangents & Chords of a Parabola
Tangent of slope m
$y = mx + \dfrac{a}{m} \quad (\text{to } y^2 = 4ax)$

Watch out for (4)

The sign of the linear term sets the direction
→ Standard Forms & Their Elements
Directrix of $y^2=4ax$ is $x=-a$ , on the OTHER side of the vertex
→ Standard Forms & Their Elements
Parabola latus rectum is $4a$ , not $2a$
→ The Latus Rectum
Focal distance is $x_1 + a$ , not $x_1 - a$
→ Focal Distance & Focal Chords

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