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, the line must satisfy the following condition:

$$ c = a/m $$

This is derived from the fact that the discriminant of the quadratic equation formed by substituting the line's equation into the parabola's equation must be zero (since there is only one point of intersection).

Let's substitute ( y = mx + c ) into ( y^2 = 4ax ):

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

Expanding and simplifying, we get:

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

This is a quadratic equation in ( x ). For the line to be tangent to the parabola, this equation must have exactly one solution for ( x ). This happens when the discriminant ( D ) is zero:

$$ D = (2mc)^2 - 4(m^2)(c^2 - 4a) = 0 $$

Simplifying, we get:

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

$$ c^2 = c^2 - 4a $$

$$ 4a = 0 $$

This is a contradiction unless ( c = a/m ), which is the condition for tangency.

Table of Differences and Important Points

Property	Line	Tangent Line to Parabola
Intersection Points	Can be 0, 1, or 2	Exactly 1 (point of tangency)
Discriminant (D)	Can be > 0, = 0, or < 0	Always = 0
Condition	No specific condition	( c = a/m )
Slope	Any value	( m = 1/(4a) \times (y_1 - c) ) where ( y_1 ) is the y-coordinate of the point of tangency

Examples

Example 1: Find the equation of the tangent line

Find the equation of the tangent line to the parabola ( y^2 = 8x ) at the point ( (2, 4) ).

Solution:

The slope of the tangent line at ( (2, 4) ) can be found using the derivative of the parabola's equation:

$$ \frac{dy}{dx} = \frac{8}{2y} $$

At the point ( (2, 4) ), the slope ( m ) is:

$$ m = \frac{8}{2 \times 4} = 1 $$

Using the point-slope form of the line equation:

$$ y - y_1 = m(x - x_1) $$

Substituting the point ( (2, 4) ) and the slope ( m = 1 ):

$$ y - 4 = 1(x - 2) $$

Simplifying:

$$ y = x + 2 $$

This is the equation of the tangent line.

Example 2: Verify the condition for tangency

Verify that the line ( y = x + 2 ) is tangent to the parabola ( y^2 = 8x ).

Solution:

Substitute ( y = x + 2 ) into ( y^2 = 8x ):

$$ (x + 2)^2 = 8x $$

Expanding:

$$ x^2 + 4x + 4 = 8x $$

Rearranging:

$$ x^2 - 4x + 4 = 0 $$

The discriminant ( D ) is:

$$ D = (-4)^2 - 4(1)(4) = 16 - 16 = 0 $$

Since ( D = 0 ), the line is tangent to the parabola.

In conclusion, understanding the condition for a line to be tangent to a parabola is essential for solving problems involving tangents and parabolas. The condition ( c = a/m ) must be satisfied, and the discriminant of the resulting quadratic equation must be zero.