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

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:

, 

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:

, 

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 – b² to 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 

Hence the required equation of the ellipse is, 

Similar Problems

Question 1. Find an equation for the ellipse lying on the y-axis with centre at (0,7), length of major axis 12 units, and length of minor axis 16 units.

Solution:

The length of major axis is, 2a = 12 => a = 6.

The length of minor axis is, 2b = 16 => b = 8.

As the ellipse lies on y-axis, the equation is 

So, the equation is .

Question 2. Find an equation for the ellipse lying on the x-axis with center at (3,2), length of major axis 2 units, and length of minor axis 8 units.

Solution:

The length of major axis is, 2a = 2 => a = 1.

The length of minor axis is, 2b = 8 => b = 4.

As the ellipse lies on x-axis, the equation is 

So, the equation is .

Question 3. Find the coordinates of the major axis of the ellipse with foci (0, 6) and minor axis (8, 0).

Solution:

We have, c = 6 and b = 8.

Put these in c² = a² – b² to find a.

a² = 6² + 8²

a² = 100

a = 10

The coordinates of major axis are (0, ±10).

Question 4. Find the equation of an ellipse centered at (7, 2) with c = 3 and b = 4.

Solution:

Given b = 4, c = 3, h = 7 and k = 2.

Put these in the formula c² = a² – b² to find a.

a² = 4² + 3²

a² = 25

a = 5

As the ellipse lies on x-axis, the equation is of the form 

So, the equation is, 

Question 5. Find the equation of the ellipse centered at the origin if a = 12, b = 6, and the minor axis is parallel to the x-axis.

Solution:

The minor axis is parallel to x-axis, so the ellipse lies on y-axis.

The equation is of the form .

Here a = 12 and b = 6, h = 0 and k = 0.

So, the equation becomes,