Condition of a Line to be Tangent to a Parabola

In geometry, a line is said to be tangent to a parabola if it touches the parabola at exactly one point. This point is known as the point of tangency. The condition for a line to be tangent to a parabola is derived from the geometric properties of the parabola and the line.

Standard Equation of a Parabola

The standard equation of a parabola with vertex at the origin and axis parallel to the y-axis is:

$$ y^2 = 4ax $$

where ( a ) is the distance from the vertex to the focus of the parabola.

Equation of a Line

The general equation of a line in the plane is:

$$ y = mx + c $$

where ( m ) is the slope of the line and ( c ) is the y-intercept.

Condition for Tangency

For a line to be tangent to a parabola, it must satisfy the condition that the discriminant of the quadratic equation formed by substituting the equation of the line into the equation of the parabola is zero. This is because a discriminant of zero implies that the quadratic equation has exactly one real root, which corresponds to the line touching the parabola at exactly one point.

Substituting ( y = mx + c ) into ( y^2 = 4ax ), we get:

$$ (mx + c)^2 = 4ax $$

Expanding and rearranging, we obtain a quadratic equation in ( x ):

$$ m^2x^2 + 2mcx + c^2 - 4ax = 0 $$

The discriminant ( D ) of this quadratic equation is:

$$ D = (2mc)^2 - 4(m^2)(c^2 - 4a) $$

For the line to be tangent to the parabola, ( D = 0 ), so:

$$ (2mc)^2 = 4(m^2)(c^2 - 4a) $$

Simplifying, we get:

$$ c^2 = 4am $$

This is the condition for a line ( y = mx + c ) to be tangent to the parabola ( y^2 = 4ax ).

Table of Differences and Important Points

Aspect	Line	Parabola
Definition	A straight path that extends infinitely in two directions	A set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix)
Equation	( y = mx + c )	( y^2 = 4ax )
Tangency Condition	( c^2 = 4am )	Not applicable
Points of Intersection	One point if tangent	Two points if secant, one if tangent, none if it does not intersect
Slope	Constant ( m )	Variable, depends on the point of tangency

Examples

Example 1: Finding the Tangent Line

Given a parabola ( y^2 = 12x ), find the equation of the tangent line at the point ( (3, 6) ).

Solution:

The slope of the tangent line at ( (x_1, y_1) ) is given by the derivative of the parabola at that point:

$$ \frac{dy}{dx} = \frac{1}{2a} \cdot 2y = \frac{y}{2a} $$

For the given parabola, ( a = 3 ), so at the point ( (3, 6) ):

$$ m = \frac{6}{2 \cdot 3} = 1 $$

The equation of the tangent line is:

$$ y = mx + c $$

We know ( m = 1 ) and the line passes through ( (3, 6) ), so:

$$ 6 = 1 \cdot 3 + c $$

Solving for ( c ), we get ( c = 3 ). Thus, the equation of the tangent line is:

$$ y = x + 3 $$

Example 2: Verifying Tangency

Verify if the line ( y = 4x + 4 ) is tangent to the parabola ( y^2 = 16x ).

Solution:

For the line to be tangent to the parabola, the condition ( c^2 = 4am ) must be satisfied. Here, ( a = 4 ) and ( m = 4 ), so:

$$ c^2 = 4 \cdot 4 \cdot 4 = 64 $$

Since ( c = 4 ), we have:

$$ 4^2 = 16 $$

The condition is not satisfied, so the line ( y = 4x + 4 ) is not tangent to the parabola ( y^2 = 16x ).

Understanding the condition for a line to be tangent to a parabola is crucial for solving problems in coordinate geometry and calculus. It is also an important concept in the study of conic sections and their applications.