Ellipses

Question

Write an equation for an ellipse with foci (0, +4) and co-vertices (+ 2, 0).

Answer 1

The foci of the ellipse (0, ± 4) and co-vertices (± 2, 0).

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.

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

Find the value of b :

Here length of the major axis = the distance between two co-vertices (2, 0) and (-2, 0).

2b = √[ (x₂ - x₁)² + (y₂ - y₁)² ]

2b = √[ (2 - (-2))² + ( 0 - 0)² ]

2b = √[4]

2b = 2

b = 1.

 

Find the center (h, k) :

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

The center of the ellipse = the midpoint of its co-vertices (0, 2) and (0, -2).

So, the center (h, k) = [ (x₁ + x₂)/2, (y₁ + y₂)/2 ] = [ (0 + 0)/2, (- 2 + 2)/2 ] = (0, 0).

 

Find the value of a :

c² = a² - b²

(4)² = a² - (1)²

16 = a² - 1

17 = a² .

 

The equation of ellipse is x²/17 + y²/1 = 1.