Understanding Tangent to a Parabola: Condition, Equation & Examples | Testbook
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A parabola, one of the most common forms of a quadratic function, is a curve where any point is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix). When a line touches this curve precisely at one point, it is referred to as a tangent to a parabola. This article aims to provide you with a comprehensive understanding of the equation of the tangent to a parabola and the point of contact of the tangent to a parabola, illustrated with examples.
Condition for Tangency
The line y = mx + c can be a tangent to the parabola y2 = 4ax if the condition c = a/m is met.
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Similarly, the line y = mx + c can be a tangent to the parabola x2 = 4ay if the condition c = -am2 is fulfilled.
The line x cos θ + y sin θ = p can be a tangent to the parabola y2 = 4ax, if the condition a sin2θ + p cos θ = 0 is satisfied.
Equation of Tangent to a Parabola
1. Point Form:
The equation of a tangent to the parabola y2 = 4ax at a specific point P(x1, y1) is given by the formula yy1 = 2a(x + x1).
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Start with the parabola:
y² = 4ax … (1) -
Since point P lies on the parabola:
y₁² = 4ax₁ … (2) -
Differentiate (1) with respect to x:
2y (dy/dx) = 4a ⇒ dy/dx = 2a/y -
At point (x₁, y₁), the slope m = 2a/y₁
-
Use the point-slope form of a line:
y – y₁ = m(x – x₁)
⇒ y – y₁ = (2a/y₁)(x – x₁)
⇒ Multiply both sides by y₁:
yy₁ – y₁² = 2a(x – x₁) -
Now plug in equation (2):
yy₁ – 4ax₁ = 2a(x – x₁)
⇒ Rearranging:
yy₁ = 2a(x + x₁)
2. Slope Form
Let’s say the slope of the tangent is m.
a) If the parabola is y² = 4ax, the tangent is:
y = mx + a/m
-
Here, the condition for a tangent is that c = a/m
-
So, you write the equation of a line as y = mx + c
-
Replace c with a/m ⇒ Final form: y = mx + a/m
-
The point where the line touches the parabola is:
(a/m², 2a/m)
b) If the parabola is x² = 4ay, the tangent is:
y = mx – am²
-
The point of contact is:
(2am, am²)
3. Parametric Form
The point on the parabola is written in terms of a parameter t as:
(at², 2at)
Then, the equation of the tangent at that point is:
ty = x + at²
-
From the point form:
yy₁ = 2a(x + x₁) -
Replace x₁ = at² and y₁ = 2at
⇒ y(2at) = 2a(x + at²)
⇒ Divide both sides by 2a:
yt = x + at²
Key Points to Know When Finding a Tangent to a Parabola
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Find the Slope (Gradient)
To get the slope of the tangent, you take the derivative of the parabola. The derivative tells you how steep the curve is at any point.
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Use the Point of Contact
You need to know the exact point on the parabola where the tangent touches. This is often given as a coordinate or through a parameter like ttt.
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Use the Slope-Point Formula
Once you know the slope and the point of contact, you can use the formula for a straight line:
y−y1=m(x−x1)
Solved Examples
Example 1:
Find the equation of the tangent to the parabola y2 = 16x at the point where the parameter t = 3.
Options:
a. x – 3y + 18 = 0
b. x – 3x – 18 = 0
c. x + 3y + 18 = 0
d. none of these
Example 2:
Determine the condition that the line x cos θ + y sin θ = P touches the parabola y2 = 4ax.
Options:
a. P cos θ + a sin2 θ = 0
b. cos θ + a P sin2 θ = 0
c. P cos θ – a sin2 θ = 0
d. None of these
Further Reading
JEE Previous Year Questions with Solutions on Parabola
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Frequently Asked Questions For Tangent to a Parabola
What is a tangent to a parabola?
A tangent to a parabola is a straight line that touches the parabola at exactly one point and has the same slope as the parabola at that point.
How is a tangent different from a secant?
A tangent touches the curve at one point, while a secant intersects it at two points.
What is the length of the tangent from an external point to a parabola?
If a point lies outside the parabola, you can use the tangent-secant theorem or derive the length using geometry or coordinate geometry methods.
Can a parabola have more than one tangent at a point?
No, a parabola can have only one unique tangent at each of its points.
What is the condition for a line to be tangent to a parabola?
Substitute the line’s equation into the parabola’s equation. If the resulting quadratic has one solution (discriminant D=0D = 0D=0), the line is tangent.
What is the locus of the point of intersection of two tangents to a parabola?
The locus of the intersection of tangents drawn from a fixed point outside the parabola is called the chord of contact, and it lies on the polar of the external point with respect to the parabola.
What is the difference between a tangent and the normal to a parabola?
A tangent touches the parabola at one point and has the same slope as the curve at that point. A normal is the line perpendicular to the tangent at the point of contact.
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