How to Find the Equation of an Ellipse with Given Foci and Major Axis Length

When dealing with ellipses, finding the equation can be a challenging yet fascinating task. This article will detail two methods to find the equation of an ellipse when given the foci and the length of the major axis. Understanding these concepts is crucial for anyone working with conic sections in mathematics.

Method 1: Using Distance Formula and Completing the Square

Let's consider an ellipse with foci S(5, 4) and S'(-1, -4), and a major axis of length 12. We aim to derive the equation of this ellipse.

Step 1: Draw a Rough Diagram

First, let's draw a rough diagram of the ellipse. The major axis is inclined at an angle of about 53 degrees with the X-axis. The center C(2, 0) is chosen such that the X-axis passes through it. The Y-axis is perpendicular to the X-axis and 2 units to the left of the center C.

Given that the length of the major axis is 2a 12, we have a 6.

Step 2: Use Distance Formula

Let P(x, y) be an arbitrary point on the ellipse. We define PS' - PS 2a. That is, PS' - PS 12.

Using the distance formula:

PS' SQRT[(x - 5)^2 (y - 4)^2]

PS SQRT[(x 1)^2 (y 4)^2]

Let Z x^2 y^2 17. Then:

PS' SQRT[Z 2x - 8y 17]

PS SQRT[Z - 24 - 8y 41]

Put Z x^2 y^2 17, then:

PS' SQRT[Z 2x - 8y]

PS SQRT[Z - 24 - 8y]

PS' - PS 12

Moving one radical to the other side:

SQRT{Z - 24 - 8y] 12 - SQRT{Z 2x - 8y}

Squaring both sides and cancelling the common term Z:

6 SQRT{Z 2x - 8y] 3x 4y 20

Squaring again and substituting Z:

27x^2 20y^2 - 24xy - 108x 48y 288 Ans.

Method 2: Using Midpoint Formula and Coordinate Rotation

Let's consider the same ellipse, but this time we'll use the midpoint formula and coordinate rotation.

Step 1: Find the Center and Slope of the Line Joining Foci

The center C of the ellipse is the midpoint of the line joining the foci S and S'. Using the midpoint formula:

C (2, 0)

The slope of the line joining S and S' is m 4/3, and the angle of inclination is t atan(4/3) 53 degrees approximately.

Step 2: Transform Coordinates with Rotation

Shift the origin to C and rotate the axes through an angle t 53 degrees. If x and y transform to X and Y in the new system, then:

X cos(t)x - 2 - sin(t)y

Y -sin(t)x - 2 cos(t)y

For S(5, 4), we get:

X 3.5 - 4.4 - 6/5 0 and Y -4.5 3.4 8/5 0

Thus, S (0, 0) in the new coordinate system. Similarly, S' (-5, 0). The first coordinate of S gives e 5/6, where e is the eccentricity.

Step 3: Derive the Equation in the New Coordinate System

The equation of the ellipse in the new system is:

X^2 / 36 Y^2 / 11 1

Replace X and Y in terms of x and y using the rotation formulas:

X 3x 4y - 6/5

Y -4x 3y 8/5

Substitute these into the equation:

(3x 4y - 6/5)^2 / 36 (-4x 3y 8/5)^2 / 11 1

After simplification, we get:

675x^2 500y^2 - 60y - 270 1200y 7200

Cancelling common factor 25, we find:

27x^2 20y^2 - 24xy - 108x 48y 288 Ans.

Conclusion

In both methods, we derived the same equation for the ellipse. Understanding these techniques is essential for those studying conic sections. The methods used include the distance formula, completing the square, midpoint formula, and coordinate rotation. These steps can be adapted to solve a wide range of problems involving ellipses.