Find the equation for the ellipse with center at (0,4), focus at (8,4), and vertex at (6, 7)

Last Updated : 15 Feb, 2023

Conic sections, commonly known as conics, are created when a plane intersects a cone. The angle at which these sections intersect determines their geometry. Conic sections are therefore classified into four types: circle, ellipse, parabola, and hyperbola. Each of these types is distinguished by its unique set of mathematical properties and equations. The ellipse is explained more below.

Ellipse

An ellipse is a conic section generated when a plane crosses a cone at an angle (β) smaller than the right angle but greater than the angle formed at the cone’s vertex (α). In other terms, an ellipse is generated when a plane slices a cone at an angle β such that α<β<90^o.

As illustrated in the image above, a cone and a plane cross at an angle that is less than the right angle but more than the angle formed at the vertex of the cone to create an ellipse as a result of 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

Find the equation for the ellipse with center at (0,4), focus at (8,4), and vertex at (6, 7).

Solution:

To find the equation of an ellipse, we need the values a and b.

The distance between the center (0, 4) and the focus (8, 4) gives c.

So, c² = (8 – 0)² + (4 – 4)²

c² = 64

The distance between the center (0, 4) and the vertex (6, 7) gives a.

So, a² = (6 – 0)² + (7 – 4)²

a² = 36 + 9

a² = 45

Put the obtained values in the formula c² = a² – b²to find b.

b² = 64 – 45

b² = 19

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

Hence the required equation of the ellipse is, $\frac{x^2}{45}+\frac{(y-4)^2}{19}=1$