Ellipse Equation?

Question

Can anybody please explain to me how to get an equation for an ellipse if I only have major axis endpoints (1, -4) and (1, 8), and manor axis length 8. Somebody please help.

Answer 1

Centre of ellipse ≡ (1, 2)

Length of semi-major axis = a = 12/2 = 6 (parallel to y-axis)
Length of semi-minor axis = b = 8/2 = 4 (parallel to x-axis)

Now for an ellipse centre (h, k) with semimajor axis parallel to y-axis is

(x - h)²/b² + (y - k)²/a² = 1

In this case

(x - 1)²/4² + (y - 2)²/6² = 1

Multiply both sides by 6² * 4² (= 24² = 576)

ie 36(x-1)² + 16(y- 2)² = 576

Divide both sides by 4 (HCF)

So 9(x-1)² + 4(y- 2)² = 144

Answer 2

The vertices of the ellipse are (1, -4) and (1, 8). Since only y varies, the ellipse is vertical. The center of the ellipse (h,k) is the midpoint of the vertices.

(h,k) = (1, 2)

The distance between the two vertices is 2a.

2a = 8 - -4 = 12
a = 6

The minor axis is 2b.

2b = 8
b = 4

The equation of the ellipse is:

(x - h)²/b² + (y - k)²/a² = 1

(x - 1)²/16 + (y - 2)²/36 = 1

Answer 3

i gotcha covered...
you have to solve for the center point of the ellipse, basically add those two points together and divide by 2 (center point or midpoint theorem). also solve for the length of the major axis using pythagorean theorem.

then...

1 = (x - h)^2/ a + (y - k)^2/b
h = x-value of center point
k = y-value of center point
a = length of minor axis (parallel to x axis)
b = length of major axis (parallel to y axis)