Write an equation for an ellipse centered at the origin, which has foci at (0,+-sqrt15) and co-vertices at (+-sqrt21,0).

Given Foci and Co-vertices: We are given the foci at (0,±√15) and co-vertices at (±√21,0). Since the foci are on the y-axis, this indicates that the ellipse is oriented vertically.Calculating c: The distance between the center and a focus is the value of c in the ellipse equation. We can calculate c using the coordinates of the foci. c = √15Calculating b: The distance between the center and a co-vertex is the value of b in the ellipse equation. We can calculate b using the coordinates of the co-vertices. b = √21Finding a: We know that for an ellipse centered at the origin, the relationship between a, b, and c is given by c² = a² - b². We can use this to find the value of a. c² = a² - b² (√15)² = a² - (√21)² 15 = a² - 21 a² = 15 + 21 a² = 36 a = √36 a = 6Writing the Equation of the Ellipse: Now that we have the values of a and b, we can write the equation of the ellipse in standard form. Since the ellipse is vertically oriented, a² will be under the y² term and b² will be under the x² term. The standard form of the ellipse is: (x - h)²/b² + (y - k)²/a² = 1 Plugging in the values for h, k, a, and b, we get: (x - 0)²/(√21)² + (y - 0)²/6² = 1 x²/21 + y²/36 = 1

Resources

Testimonials

Plans

Resources

Testimonials

Plans

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,+-sqrt15) and co-vertices at
(+-sqrt21,0).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm \sqrt{15})$ and co-vertices at $( \pm \sqrt{21}, 0)$ .

View step-by-step help

Home

Math Problems

Algebra 2

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 \sqrt{15})

and co-vertices at

( \pm \sqrt{21}, 0)

Given Foci and Co-vertices: We are given the foci at $(0,\pm\sqrt{15})$ and co-vertices at $(\pm\sqrt{21},0)$ . Since the foci are on the y-axis, this indicates that the ellipse is oriented vertically.
Calculating $c$ : The distance between the center and a focus is the value of $c$ in the ellipse equation. We can calculate $c$ using the coordinates of the foci. $c = \sqrt{15}$
Calculating $b$ : The distance between the center and a co-vertex is the value of $b$ in the ellipse equation. We can calculate $b$ using the coordinates of the co-vertices. $b = \sqrt{21}$
Finding a: We know that for an ellipse centered at the origin, the relationship between $a$ , $b$ , and $c$ is given by $c^2 = a^2 - b^2$ . We can use this to find the value of $a$ .
$c^2 = a^2 - b^2$
$(\sqrt{15})^2 = a^2 - (\sqrt{21})^2$
$15 = a^2 - 21$
$a^2 = 15 + 21$
$a^2 = 36$
$b$ $0$
$b$ $1$
Writing the Equation of the Ellipse: Now that we have the values of $a$ and $b$ , we can write the equation of the ellipse in standard form. Since the ellipse is vertically oriented, $a^2$ will be under the $y^2$ term and $b^2$ will be under the $x^2$ term. $\newline$ The standard form of the ellipse is: $\newline$ $(x - h)^2/b^2 + (y - k)^2/a^2 = 1$ $\newline$ Plugging in the values for $h$ , $k$ , $a$ , and $b$ , we get: $\newline$ $b$ $1$ $\newline$ $b$ $2$