Parabola: Tangents, Normals and Focal Chords | NIMCET

Quick answer: For $y^2 = 4ax$ , the tangent at $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$ , and at the point $t$ it is $ty = x + at^2$ . The line $y = mx + c$ is a tangent when $c = \frac{a}{m}$ , so the slope form is $y = mx + \frac{a}{m}$ . The normal at $t$ is $y = -tx + 2at + at^3$ . A chord joining $t_1$ and $t_2$ passes through the focus exactly when $t_1t_2 = -1$ .

This continues from parabola focus, directrix and standard forms.

Tangent

Form	Tangent to $y^2 = 4ax$
Point $(x_1, y_1)$	$yy_1 = 2a(x + x_1)$
Parametric $t$	$ty = x + at^2$
Slope m	$y = mx + \frac{a}{m}$ , touching at $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$

Condition of tangency: $y = mx + c$ touches $y^2 = 4ax$ if $c = \frac{a}{m}$ .

For $x^2 = 4ay$ , the condition is $c = -am^2$ .

Normal

Form	Normal to $y^2 = 4ax$
Point $(x_1, y_1)$	$y - y_1 = -\frac{y_1}{2a}(x - x_1)$
Parametric $t$	$y = -tx + 2at + at^3$
Slope m	$y = mx - 2am - am^3$ , at the point $(am^2, -2am)$

Three normals: a cubic in m shows that up to three normals can be drawn from a point to a parabola. The sum of their slopes is 0.

If the normal at $t_1$ meets the parabola again at $t_2$ , then

$t_2 = -t_1 - \frac{2}{t_1}$

Chords

Chord	Equation or property
Joining $t_1$ and $t_2$	$2x - (t_1 + t_2)y + 2at_1t_2 = 0$
Focal chord	$t_1t_2 = -1$
Chord of contact from $(x_1, y_1)$	$yy_1 = 2a(x + x_1)$
Chord with midpoint $(x_1, y_1)$	$T = S_1$

Properties worth remembering

Property	Statement
Tangents at ends of a focal chord	Perpendicular, and meet on the directrix
Intersection of tangents at $t_1, t_2$	$(at_1t_2, a(t_1 + t_2))$
Perpendicular tangents	Meet on the directrix $x = -a$
Focal chord length	$a\left(t + \frac{1}{t}\right)^2$
Harmonic property	$\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$ for a focal chord PQ
Reflection property	Rays parallel to the axis reflect through the focus

Worked examples

Example 1: tangent at a point

Find the tangent to $y^2 = 8x$ at $(2, 4)$ .

Here $a = 2$ : $4y = 4(x + 2)$ , so $y = x + 2$ .

Example 2: tangency condition

Find c so that $y = 3x + c$ touches $y^2 = 12x$ .

$a = 3$ , $m = 3$ , so $c = \frac{3}{3} = 1$ .

Example 3: focal chord

One end of a focal chord of $y^2 = 4x$ is $(4, 4)$ . Find the other end.

Step 1: $a = 1$ ; $(4, 4) = (t^2, 2t)$ gives $t_1 = 2$ .

Step 2: $t_2 = -\frac{1}{2}$ , so the other end is $\left(\frac{1}{4}, -1\right)$ .

Check: $\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{5} + \frac{1}{1.25} = 0.2 + 0.8 = 1 = \frac{1}{a}$ ✓.

Example 4: normal at a parameter

Find the normal to $y^2 = 4x$ at $t = 1$ .

$y = -x + 2 + 1$ , so $x + y = 3$ . The point is $(1, 2)$ , and $1 + 2 = 3$ ✓.

Summary table

Result	Formula for $y^2 = 4ax$
Tangent at t	$ty = x + at^2$
Slope-form tangent	$y = mx + \frac{a}{m}$
Normal at t	$y = -tx + 2at + at^3$
Focal chord	$t_1t_2 = -1$
Tangent intersection	$(at_1t_2, a(t_1 + t_2))$

Exam tips

Convert points on the parabola to parameters; the algebra becomes much shorter.
The harmonic property gives the semi-latus rectum: $\frac{2 \cdot SP \cdot SQ}{SP + SQ} = 2a$ .
For "perpendicular tangents meet on…", the answer is the directrix.

Next, see ellipse: eccentricity, foci and tangents, or browse the NIMCET Coordinate Geometry and Vectors chapter.

Common mistakes

Writing the tangency condition as $c = am$ instead of $\frac{a}{m}$ .
Using $t_1t_2 = 1$ for a focal chord.
Mixing up the tangent and normal parametric forms.
Forgetting that a normal from a general point can give up to three solutions.

Frequently Asked Questions

What is the equation of the tangent to y² = 4ax at (x₁, y₁)?

yy₁ = 2a(x + x₁).

What is the condition for y = mx + c to touch y² = 4ax?

c = a/m.

What is the condition for a focal chord of a parabola?

If the ends have parameters t₁ and t₂, then t₁t₂ = −1.

What is the normal to y² = 4ax at the point t?

y = −tx + 2at + at³.

Where do perpendicular tangents to a parabola meet?

They meet on the directrix.