How to Derive the Equation of an Ellipse Given Its Foci

Understanding the properties of an ellipse, or ldquo;the locus of points where the sum of the distances from two fixed points (the foci) to any point on the ellipse is constant, is crucial in both mathematics and real-world applications.

Introduction to Ellipse and Foci

The ellipse is a conic section with a fascinating and practical definition: it is the set of all points in a plane where the sum of the distances from two fixed points (the foci) to any point on the ellipse is a constant. The centers, foci, vertices, and semi-major and semi-minor axes are all key elements in the definition and derivation of the ellipse equation. This article will focus on the step-by-step process of determining the equation of an ellipse given its foci.

Step-by-Step Derivation of the Ellipse Equation

Let's illustrate this process with an example where we want to derive the equation of an ellipse with foci at the points (2, -2) and (4, -2).

Step 1: Determine the Center of the Ellipse

The center of the ellipse is the midpoint of the line segment connecting the two foci. Using the midpoint formula:

$$ h frac{x_1 x_2}{2}, quad k frac{y_1 y_2}{2} $$

Given the foci (2, -2) and (4, -2), we calculate the center as:

$$ h frac{2 4}{2} 3, quad k frac{-2 - 2}{2} -2 $$

Thus, the center of the ellipse is (3, -2).

Step 2: Determine the Distance Between the Foci

The distance c between the foci is calculated using the distance formula:

$$ c frac{d}{2} frac{sqrt{(4-2)^2 (-2 2)^2}}{2} 1 $$

This means the distance from the center to each focus is 1.

Step 3: Determine the Semi-Major and Semi-Minor Axes

The standard form of the ellipse equation is given by:

$$ frac{(x - h)^2}{a^2} frac{(y - k)^2}{b^2} 1 $$

Here, (c^2 a^2 - b^2). Since the foci are horizontally aligned, the major axis is horizontal. We need to determine (a) (semi-major axis) and (b) (semi-minor axis).

Assuming (a > c), let's denote (a b c). Using (c 1), we choose a reasonable value for (a). For example, if we let (a 2), then:

$$ c^2 a^2 - b^2 Rightarrow 1^2 2^2 - b^2 Rightarrow 1 4 - b^2 Rightarrow b^2 3 $$

Therefore, (b sqrt{3}).

Step 4: Write the Equation of the Ellipse

Substituting (h), (k), (a^2), and (b^2) into the standard form, we get:

$$ frac{(x - 3)^2}{4} frac{(y 2)^2}{3} 1 $$

Thus, the equation of the ellipse with foci at (2, -2) and (4, -2) is:

$$ frac{(x - 3)^2}{4} frac{(y 2)^2}{3} 1 $$

Additional Information

Equation of an ellipse is often derived given the coordinates of one point on the ellipse. The standard form of an ellipse in Cartesian coordinates with its center at the origin and major axis along the x-axis is:

The foci are at ((pm c, 0)) and the vertices are at ((pm a, 0)). The distance from a point ((x, y)) to each focus is represented by (sqrt{(x - c)^2 y^2}) and (sqrt{(x c)^2 y^2}). The point ((x, y)) lies on the ellipse if the sum of these distances is equal to (2a).

The equation of the ellipse in standard form is:

$$ frac{x^2}{a^2} frac{y^2}{b^2} 1 $$

Conclusion

Carefully following the steps outlined, one can derive the equation of an ellipse given its foci. This process not only simplifies the mathematical representation but also provides a deeper understanding of the geometric properties and relationships within the ellipse. Understanding these derivation steps is fundamental for applications in fields like astronomy, engineering, and architecture.

For further reading, explore the article on Ellipse - Wikipedia for a comprehensive overview and more examples.