ellipse equation

Question

How do I find the equation of an ellipse if I have the major axis endpoints and the minor axis length?

End points  (1,-4) and (1,8)

minor axis length 8

Answer 1

given that  and minor axis length is 8

Apply formula:

Substitute a,b  values  in the  given equation

The ellipse equation is:

Comments

The above solution is wrong,

See the correct solution, answered by steve.

Answer 2

The standard form of the equation of an ellipse center (h, k) with major and minor axes of lengths 2a and 2b (where 0 < b < a) is (x - h)²/a² + (y - k)²/b² = 1(major axis is horizontal) or (x - h)²/b² + (y - k)²/a² = 1(major axis is vertical).

The vertices and foci lie on the major axis, a and c units, respectively, from the center (h, k) and the relation between a, b and c is c² = a² - b².

The end points of the major axis are (1, - 4) and (1, 8), and length of the minor axis is 8 units.

To find the equation of ellipse, first find major axis is vertical or horizontal, and next find the values of a, b, h and k.

The end points of the major axis means that these are vertices of the ellipse.

Since the x - coordinate is same in the vertices, the major axis is vertical, then the equation of ellipse is (x - h)²/b² + (y - k)²/a² = 1.

Minor axis length = 2b = 8 ⇒ b = 4.

Find the value of a :

Here length of the major axis = the distance between two vertices points (1, - 4) and (1, 8).

2a = √[ (x₂ - x₁)² + (y₂ - y₁)² ]

2a = √[ (1 - 1)² + ( 8 - (- 4) )² ]

2a = √[ (0)² + (12)² ]

2a = 12

a = 6.

Find the center (h, k) :

The center of the ellipse lies at the midpoint of its vertices or foci.

The center of the ellipse = the midpoint of its vertices (1, - 4) and (1, 8).

So, the center (h, k) = [ (x₁ + x₂)/2, (y₁ + y₂)/2 ] = [ (1 + 1)/2, (- 4 + 8)/2 ] = (1, 2).

The equation of ellipse is (x - 1)²/4² + (y - 2)²/6² = 1.