Condition of a Line to be Tangent to a Parabola

A parabola is a conic section that can be described by a quadratic equation in two variables. The standard form of a parabola with its vertex at the origin and axis of symmetry along the y-axis is:

$$ y^2 = 4ax $$

where (a) is a constant that determines the width and direction of the parabola.

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 that it must satisfy the parabola's equation at exactly one point, and its slope must be equal to the derivative of the parabola's equation at that point.

Condition for Tangency

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.

For the line to be tangent to the parabola (y^2 = 4ax), it must satisfy the following condition:

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

This equation must have exactly one solution for (x), which means the discriminant of the quadratic equation must be zero. The discriminant (D) of a quadratic equation (ax^2 + bx + c = 0) is given by:

$$ D = b^2 - 4ac $$

For the tangency condition, we have:

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

Solving for (c), we get:

$$ c = \pm 2a/m $$

This gives us the condition for the line (y = mx + c) to be tangent to the parabola (y^2 = 4ax):

$$ c = \pm \frac{2a}{m} $$

Table of Differences and Important Points

Aspect	Parabola (y^2 = 4ax)	Line (y = mx + c)	Tangency Condition
Equation	Quadratic	Linear	(c = \pm \frac{2a}{m})
Variables	(x, y)	(x, y)	(m, c)
Parameters	(a) (width/direction)	(m) (slope), (c) (y-intercept)	(a) (from parabola), (m) (from line)
Graph	U-shaped curve	Straight line	Line touches parabola at one point
Discriminant	Not applicable	Not applicable	Must be zero for tangency

Examples

Example 1: Find the equation of the tangent line

Find the equation of the tangent line to the parabola (y^2 = 8x) with a slope of 2.

Solution:

Given the slope (m = 2) and the parabola (y^2 = 8x) (where (a = 2)), we use the condition for tangency:

$$ c = \pm \frac{2a}{m} = \pm \frac{2 \cdot 2}{2} = \pm 2 $$

Therefore, the equations of the tangent lines are:

$$ y = 2x + 2 \quad \text{and} \quad y = 2x - 2 $$

Example 2: Verify the condition of tangency

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

Solution:

For the parabola (y^2 = 16x), we have (a = 4). The given line has a slope (m = 4) and y-intercept (c = -4).

Using the condition for tangency:

$$ c = \pm \frac{2a}{m} = \pm \frac{2 \cdot 4}{4} = \pm 2 $$

Since the given y-intercept is (-4), which does not equal (\pm 2), we might initially think that the line is not tangent. However, we must check the discriminant of the combined equation:

$$ (4x - 4)^2 = 16x $$

Expanding and simplifying:

$$ 16x^2 - 32x + 16 = 16x $$

$$ 16x^2 - 32x + 16 - 16x = 0 $$

$$ 16x^2 - 48x + 16 = 0 $$

Dividing by 16:

$$ x^2 - 3x + 1 = 0 $$

The discriminant of this quadratic equation is:

$$ D = (-3)^2 - 4(1)(1) = 9 - 4 = 5 $$

Since the discriminant is not zero, the line (y = 4x - 4) is not tangent to the parabola (y^2 = 16x). This confirms our initial suspicion based on the condition for tangency.

Understanding the condition for a line to be tangent to a parabola is crucial for solving problems related to tangents and normals in the study of conic sections. It is important to remember that the discriminant must be zero for tangency, and the slope of the tangent line must match the derivative of the parabola at the point of tangency.