Question

Write the equation of the ellipse and find the foci with the given vertices (2,3),(2,13),(0,8),(4,8)

Answer 1

Explanation:

Equation of ellipse depends on the points

The points that lie on the horizontal axis is (0,8) and (4,8). They have a distance of 4, therefore a radius of 2

The points that lie on the vertical axis is (2,3) and (2,13)

They have a distance of 10, therefore a radius of 5

Since the vertical axis is bigger, we have a vertical ellipse which is the equation of,

`\frac{(y - k) {}^(2) }{ {a}^(2) } + \frac{(x - h) {}^(2) }{ {b}^(2) } = 1`

Where a is the major radius

b is the minor radius

(h,k) is the center

Step 1: Finding the center

The center should be the midpoint of the horizontal vertices or vertical vertices

Let's use the horizontal vertices

`(((0 + 4)/(2) ) = 2`

`( (8 + 8)/(2) ) = 8`

So the center is (2,8)

Step 2: Finding the major and minor radius

We already know it , a is 5

b is 2, so we as of now, have this equation

`\frac{(y -8) {}^(2) }{25} + \frac{(x - 2) {}^(2) }{4} = 1`

This is the equation of the ellisoe

Step 3: Find the foci

The foci for a vertical ellipse should lie on the same line as the same line as the major vertices and center.

The equation for an ellipse foci that is vertical are

(x,y+c) and (x,y-c)

Where C is

`c = \sqrt{ {a}^(2) - {b}^(2) }`

`c = √(25 - 4)`

`c = √(21)`

So our foci are

,,,

`(2,8 + √(21) )`

and

`(2,8 - √(21) )`

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