Introduction to the Problem

The problem of finding the equation of a common tangent between two parabolic curves can be approached using principles of calculus and algebra. In this article, we will explore the process to determine the common tangent for the parabolas given by y^2 2x and x^2 16y. This involves understanding the slopes of tangents at various points on these curves and setting up equations to find their intersection.

Understanding the Parabolas

Let's start with a brief overview of the two parabolas:

y^2 2x is a parabola that opens to the right. x^2 16y is a parabola that opens upwards.

Our goal is to identify the common tangent - a line that touches both parabolas at exactly one point each, making it a common line of tangency.

Step-by-Step Calculation of Tangents

The first step in solving this problem involves finding the slopes of the tangents at various points on each parabola. This is achieved through implicit differentiation.

1. Tangent to the Parabola y^2 2x

Solve for frac{dy}{dx} using implicit differentiation: 2y frac{dy}{dx} 2 implies frac{dy}{dx} frac{1}{y} The slope of the tangent line at a point (x1, y1) on this curve is frac{1}{y_1}.

2. Tangent to the Parabola x^2 16y

Solve for frac{dy}{dx} using implicit differentiation: 2x 16 frac{dy}{dx} implies frac{dy}{dx} frac{x}{8} The slope of the tangent line at a point (x2, y2) on this curve is frac{x_2}{8}.

To find the common tangent, we need to set the slopes equal to each other:

frac{1}{y_1} frac{x_2}{8}

Equations of the Tangents

Now let's derive the equations of the tangent lines at the points of tangency on each parabola.

1. Tangent to y^2 2x at (x1, y1)

The equation of the tangent line is:

y - y_1 frac{1}{y_1}x - x_1

Rearranging gives:

yy_1 - y_1^2 x - x_1 implies x - yy_1 - y_1^2 x_1

2. Tangent to x^2 16y at (x2, y2)

The equation of the tangent line is:

y - y_2 frac{x_2}{8}x - x_2

Rearranging gives:

8y - y_2 cdot 8 x cdot x_2 - x_2^2 implies x - 8y - 8y_2 x_2^2

To find the common tangent, we assume the form of the tangent line to be y mx c.

Deriving the Conditions for a Common Tangent

1. For y^2 2x

Substitute y mx c into y^2 2x:

m^2x^2 - 2mcx - c^2 - 2x 0

The discriminant of this quadratic equation must be zero for it to have a double root, leading to the condition:

4mc - 1^2 - 4m^2c^2 0 implies mc - 1^2 m^2c^2

2. For x^2 16y

Substitute y mx c into x^2 16y:

x^2 16mx 16c implies x^2 - 16mx - 16c 0

The discriminant of this quadratic equation must also be zero, leading to:

256m^2 - 64c 0 implies 256m^2 64c

Solving the System of Equations

By solving the equations derived from the conditions, we typically find that the common tangents are:

y x - 4 quad text{and} quad y -frac{1}{4}x - 4

These represent the equations of the common tangents to the given parabolas y^2 2x and x^2 16y.