Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Write an equation for an ellipse centered at the origin, which has foci at
(0,+-15) and vertices at
(0,+-25).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm 15)$ and vertices at $(0, \pm 25)$ .

View step-by-step help

View step-by-step help

Write equations of ellipses in standard form using properties

Full solution

Q. Write an equation for an ellipse centered at the origin, which has foci at

(0, \pm 15)

and vertices at

(0, \pm 25)

.

Identify coordinates of foci and vertices: Identify the coordinates of the foci and vertices. $\newline$ Foci: $(0, \pm 15)$ $\newline$ Vertices: $(0, \pm 25)$ $\newline$ Since both foci and vertices lie on the y-axis, the ellipse is vertical.
Determine values of $a$ and $c$ : Determine the values of $a$ and $c$ . The distance from the center to a vertex is ' $a$ ', and the distance from the center to a focus is ' $c$ '. $a = 25$ (since the vertices are at $(0, \pm 25)$ ) $c = 15$ (since the foci are at $(0, \pm 15)$ )
Calculate value of b: Calculate the value of b using the relationship $c^2 = a^2 - b^2$ . $\newline$ $c^2 = a^2 - b^2$ $\newline$ $15^2 = 25^2 - b^2$ $\newline$ $225 = 625 - b^2$ $\newline$ $b^2 = 625 - 225$ $\newline$ $b^2 = 400$ $\newline$ $b = \sqrt{400}$ $\newline$ $b = 20$
Write equation in standard form: Write the equation of the ellipse in standard form. $\newline$ Since the ellipse is vertical, the standard form of the equation is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ . $\newline$ Substitute the values of a and b: $\newline$ $\frac{x^2}{20^2} + \frac{y^2}{25^2} = 1$ $\newline$ $\frac{x^2}{400} + \frac{y^2}{625} = 1$
Simplify equation if necessary: Simplify the equation if necessary. $\newline$ The equation is already in its simplest form, so no further simplification is needed. $\newline$ Final equation: $\frac{x^2}{400} + \frac{y^2}{625} = 1$

More problems from Write equations of ellipses in standard form using properties

Question

Write the equation in standard form for the ellipse with center at the origin, vertex

(-10,0)

, and co-vertex

(0,3)

Posted 1 month ago

Question

Write the equation in standard form for the ellipse

7x^2 + y^2 = 21

\newline

Posted 2 months ago

Question

What is the center of the ellipse

x^2 + 2y^2 - 16 = 0

\newline

Write your answer in simplified, rationalized form.

\newline

(\_\_\_\_\_,\_\_\_\_\_)

Posted 1 month ago

Question

What is the length of the major axis of the ellipse

\frac{x^2}{9} + \frac{y^2}{2} = 1

\newline

Write your answer in simplified, rationalized form.

\newline

\_\_\_\_\_

Posted 1 month ago

Question

What is the center of the ellipse

\left(\frac{(x - 5)^2}{9}\right) + \left(\frac{(y - 2)^2}{54}\right) = 1

\newline

Write your answer in simplified, rationalized form.

\newline

(________ , _______)

Posted 2 months ago

Question

The equation of an ellipse is given below.

\newline

\frac{(x+15)^{2}}{676}+\frac{(y-4)^{2}}{100}=1

\newline

What are the foci of this ellipse?

\newline

1

\newline

(-15+\sqrt{24}, 4)

(-15-\sqrt{24}, 4)

\newline

(-15,4+\sqrt{24})

(-15,4-\sqrt{24})

\newline

(-39,4)

(9,4)

\newline

(-15,28)

(-15,-20)

Posted 3 months ago

Question

The equation of an ellipse is given below.

\newline

\frac{(x-5)^{2}}{3}+\frac{(y-7)^{2}}{6}=1

\newline

What are the foci of this ellipse?

\newline

1

\newline

(5,7+\sqrt{3})

(5,7-\sqrt{3})

\newline

(-5,-7+\sqrt{3})

(-5,-7-\sqrt{3})

\newline

(-5+\sqrt{3},-7)

(-5-\sqrt{3},-7)

\newline

(5+\sqrt{3}, 7)

(5-\sqrt{3}, 7)

Posted 3 months ago

Question

The equation of an ellipse is given below.

\newline

\frac{x^{2}}{46}+\frac{(y+8)^{2}}{26}=1

\newline

What are the foci of this ellipse?

\newline

1

\newline

(0+\sqrt{20}, 8)

(0-\sqrt{20}, 8)

\newline

(0,8+\sqrt{20})

(0,8-\sqrt{20})

\newline

(0,-8+\sqrt{20})

(0,-8-\sqrt{20})

\newline

(0+\sqrt{20},-8)

(0-\sqrt{20},-8)

Posted 3 months ago

Question

Write an equation for an ellipse centered at the origin, which has foci at

( \pm 12,0)

and vertices at

( \pm 13,0)

Posted 3 months ago

Question

Write an equation for an ellipse centered at the origin, which has foci at

(0, \pm 6)

and vertices at

(0, \pm \sqrt{37})

Posted 3 months ago