Find the standard form of the equation of the ellipse satisf

Find the standard form of the equation of the ellipse satisfying the given conditions. Endpoints of major axis (- 4, 7) and (- 4, 1) Endpoints of minor axis: (- 2, 4) and (- 6, 4) Standard form of the equation

Solution

Solution:

The standard form is

(x-h)²/a² + (y-k)²/b² = 1

(h,k) is the center

a = The distance from the center to either end of the major axis

b =  The distance from the center to either end of the minor axis

The distance between the endpoints of the major axis is (7- 1) = 6.

So, the length of the semi-major axis is (6/2) = 3.

Similarly, the distance between the endpoints of the minor axis is (-2- (-6)) = 4.

So, the length of the semi-minor axis is (4/2) = 2.

The centre of the ellipse is located at the coordinates (-4,4).

That the general equation for an ellipse centered at the coordinates (h,k) is

(x - h)² / a² + (y - k)²/ b² = 1,

where a and b are the lengths of the semi-major and semi-minor axes.

Thus, the standard form of the ellipse is

(x + 4)² / (4) + (y - 4)² / (9) = 1