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nicekikah

Answered question

2021-08-13

Find an equation of the conic described.Graph the equation.
Ellipse; center at (0,0); focus at (0,3); vertex at (0, 5)

Derrick

Skilled2021-08-14Added 94 answers

The solution is written below

Vasquez

Expert2021-12-28Added 669 answers

Step 1

Consider the given information about conic Ellipse.

Center: (0,0)

Focus: (0,3)

Vertex: (0,5)

The major axis for the ellipse is y and minor axis is x that is, a>b

So, the equation of the ellipse is,

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

Where, (h,k) is the center of the ellipse

Compare the coordinate of vertex with $(0, \pm a)$

a=5

Compare the coordinate of focus with $(0, \pm c)$ where,

$c^{2} = a^{2} - b^{2}$

c=3

Substitute c=3 and a=5 into $c^{2} = a^{2} - b^{2}$

$3^{2} = 5^{2} - b^{2}$

$b = \sqrt{25 - 9}$

b=4

Step 2

Substitute the value center, a and b into equation of ellipse

$\frac{(x - 0)^{2}}{4^{2}} + \frac{(y - 0)^{2}}{5^{2}} = 1$

$\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1$

therefore, the equation of the ellipse is $\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1$

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