How to Find the Equation of a Parabola Passing Through Two Given Points and Having Its Vertex on a Specified Line

In this article, we will explore a systematic method to find the equation of a parabola that passes through two given points and has its vertex on a specified line. This is a common problem in geometry and has applications in various fields such as physics, engineering, and computer science.

Understanding the Parabolic Equation

The general form of a parabola in vertex form is given by:

y a(x - h)2 k

where (h, k) is the vertex of the parabola, and a determines the direction and width of the parabola.

Identifying the Given Information

To solve this problem, you need to identify the following:

Two points through which the parabola passes. A line on which the vertex of the parabola lies.

Let the points be P1(x1, y1) and P2(x2, y2). The line on which the vertex lies is expressed as y mx b.

Expressing the Vertex

Since the vertex (h, k) lies on the given line, we can express k in terms of h using the line equation:

k mh b

Substituting Points into the Parabola Equation

Substitute the coordinates of the given points into the vertex form to obtain two equations:

y1 a(x1 - h)2 k

y2 a(x2 - h)2 k

Rearranging these equations:

k y1 - a(x1 - h)2

k y2 - a(x2 - h)2

Setting the two expressions for k equal to each other gives:

y1 - a(x1 - h)2 y2 - a(x2 - h)2

Simplifying and Solving for a and h

This equation can be simplified to solve for a in terms of h:

a(x2 - h)2 - (x1 - h)2 y2 - y1

Isolating a:

a frac{y2 - y1 (x1 - h)2 - (x2 - h)2}{(x2 - h)2 - (x1 - h)2}

Substituting Back to Find k

Once you find h and a, substitute h back into the expression for k:

k mh b

Writing the Final Equation

Finally, substitute h, k, and a back into the vertex form of the equation:

y a(x - h)2 k

Example

Suppose we want the parabola to pass through points (1, 2) and (3, 4) with the vertex on the line y 2x 1.

Let the vertex be (h, k 2h 1).

Substitute the points:

2 a(1 - h)2 2h 1

4 a(3 - h)2 2h 1

Solve these two equations to find a and h.

Note: This systematic approach allows you to find the equation of the parabola that meets the given criteria.