Condition for Line y mx c to be Tangent to Parabola y^2 4ax

Introduction: A tangent line to a curve, such as a parabola, touches the curve at exactly one point. By understanding the conditions under which a line y mx c is tangent to a specific parabola, such as y^2 4ax, we gain insight into the nature of tangency and mathematical analysis.

Understanding the Parabola

The equation y^2 4ax represents a parabola that opens to the right when a 0. This parabola has its vertex at the origin and the axis of symmetry along the x-axis. To determine when the line y mx c is tangent to this parabola, we need to analyze the intersection points.

General Method for Tangency

A tangent line to a curve y f(x) at a point (x_0, f(x_0)) is given by the equation:

y f'(x_0)(x - x_0) f(x_0)

Here, f'(x_0) is the derivative of f evaluated at x_0. For the parabola y^2 4ax, we first express y as a function of x and then find the derivative.

Expressing the Parabola as a Function

Given y^2 4ax, we can express y as:

y ±2√a√x

For simplicity, let's consider the positive branch:

y 2√a√x

Finding the Derivative

We need the derivative of y 2√a√x to find the slope of the tangent line at any point:

y' (2√a)(1/2)(1/√x) √a/√x

Conditions for Tangency

Let's assume the line y mx c is tangent to the parabola at some point x b. At this point, the slope of the line must match the slope of the parabola.

m √a/√b

Solving for c using the point of tangency:

c 2√a√b - mb 2√a√b - √a/√b * 2√a√b 2√a√b - 2a/√b

Generalizing the Solution

Therefore, the condition for the line y mx c to be tangent to the parabola y^2 4ax is:

m √a/√b c 2√a√b - 2a/√b

Special Cases

For a different branch, if the parabola is defined as y -2√a√x, the process is similar, but the sign of the coefficient in the derivative changes:

m -√a/√b

And solving for c similarly:

c -2√a√b 2a/√b

Conclusion

The detailed derivation and general conditions discussed provide a comprehensive understanding of how to determine if a line is tangent to a given parabola. This knowledge is pivotal in calculus, geometry, and applied mathematics, offering insights into the nature of tangency and the behavior of curves.