Question

Find an equation for the given ellipse. Graph the equation.

vertices at (4, 3) and (4, 9);    focus at (4, 8)

Answer 1

Step 1:

The focus of the ellipse is .

The vertices of the ellipse are ,

The -coordinate of vertices and focus are same, so the ellipse is vertical.

The standard form of vertical ellipse is .

where is the center of the ellipse, is the length of the semi minor axis and is the length of the semi major axis.

The center is a mid point of two vertices.

The distance between center and vertex is .

The distance between center and focus is .

Step 2:

The mid point of and is center of ellipse

The center of ellipse .

The length of major axis is the distance between center and vertex.

is the distance between center and focus .

Now .

Substitute the values of , and in standard form of ellipse.

The equation of ellipse is .

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Step 3:

Graph the ellipse.

Graph the equation .

Plot the center point .

Plot the focus points and .

Plot the vertex points  , .

Sketch the ellipse by using semi major and semi minor axis lengths.

Solution :

The equation of ellipse is .