equation for ellipse

Question

center (0,0), height 2 units, width 10 units.

Answer 1

Standard equation of ellipse is (x-h)^2/a^2+(y-k)^2/b^2 = 1

Here (h,k) = (0,0)

Height = 2 units = b

Width = 10 units = a

Substitue the values in  standard form.

(x-0)^2/2^2+(y-0)^2/10^2 = 1

x^2/4+y^2/100 = 1

equation of ellipse is x^2/4+y^2/100 = 1.

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 center is (0, 0), height is 2 units and width is 10 units.

Here 0 < height < width (0 < 2 < 10), so width is taken as major axis.

Width = length of the major axis = 2a = 10  ⇒  a = 5.

Height = length of the minor axis = 2b = 2  ⇒  b = 1.

• Case 1 :

By height and width I assume they mean the length and width of horizontal and vertical lines that would enclose the ellipse within a rectangle. I assume this because, otherwise, they would have asked for the major and minor axis.

The standard form of the equation of an ellipse (major axis is vertical) is (x - h)²/b² + (y - k)²/a² = 1.

(x - 0)²/1² + (y - 0)²/5² = 1

x² + y²/25 = 1

25x² + y² = 25.

• Case 2 :

By height and width I assume they mean the length and width of vertical and horizontal lines that would enclose the ellipse within a rectangle. I assume this because, otherwise, they would have asked for the major and minor axis.

The standard form of the equation of an ellipse (major axis is horizontal) is (x - h)²/a² + (y - k)²/b² = 1.

(x - 0)²/5² + (y - 0)²/1² = 1

x²/25 + y² = 1

x² + 25y² = 25.

The equation of ellipse is 25x² + y² = 25 or x² + 25y² = 25.