How to find the equation of an ellipse with foci and points?

Last Updated : 06 Mar, 2023

When a plane crosses a cone, conic sections, also known as conics, are created. The geometry of these sections is determined by the angle at which they cross. As a result, conic sections are divided into four categories: circle, ellipse, parabola, and hyperbola. Each of these forms has its own set of mathematical features and equations. The ellipse is discussed below.

Ellipse

As a conic section, an ellipse is a shape formed when a plane intersects a cone at an angle (β) less than the right angle but more than the angle formed at the vertex of a cone (α). In other words, an ellipse is formed when a plane cuts a cone at an angle β such that α<β<90^o.

As shown in the figure above, a cone and a plane intersect at an angle β which is less than the right angle but more than α to form an ellipse due to the intersection.

Equation of an ellipse

The standard equation of an ellipse centered at (h, k) with a major axis parallel to the x-axis is given by:

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

where the coordinates of the vertex are (h±a, 0), coordinates of co-vertex are (h, k±b) and the coordinates of foci are (h±c, k), where c² = a² – b².

Horizontal ellipse

The standard equation of an ellipse centered at (h, k) with a major axis parallel to the y-axis is given by:

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

where the coordinates of the vertex are (h, k±a), coordinates of co-vertex are (h±b, k) and the coordinates of foci are (h, k±c), where c² = a² – b².

Vertical ellipse

How to find the equation of an ellipse with foci and points?

Solution:

To find the equation of an ellipse, we need the values a and b. Now, it is known that the sum of the distances of a point lying on an ellipse from its foci is equal to the length of its major axis, 2a. The value of a can be calculated by this property. To calculate b, use the formula c² = a² – b². Substitute the obtained values of a and b in the standard form to get the required equation.

Let us understand this method in more detail through an example.

Example: Say, an ellipse passing through the origin with foci (±4, 0) and point (–4, 1.8).

Using the formula, we have

$2a=1.8+\sqrt{(4-(-4))^2+(0-1.8)^2}$

2a = 10

a = 5

Put a = 5 in c² = a² – b² to find b.

b² = 25 – 16

b² = 9

As the ellipse lies on x-axis, the equation is of the form $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ .

So, the equation is, $\frac{x^2}{25}+\frac{y^2}{9}=1$ .