Question

Find the center, vertices, and foci of the ellipse with equation 3x2 + 8y2 = 24

Answer 1

Centre: (0,0)

Vertices: (2sqrt(2) , 0) (-2sqrt(2) , 0)

Co-vertices: (0, sqrt(3)) (0, -sqrt(3))

Foci: (sqrt(5), 0) (-sqrt(5) , 0)

Explanation:

(x - h)²/a² + (y - k)²/b² = 1

3x²/24 + 8y²/24 = 1

x²/8 + y²/3 = 1

Centre: (0,0)

Vertices:

y² = 3

y = +/- sqrt(3)

(0, sqrt(3))

(0, -sqrt(3))

x² = 8

x = +/- sqrt(8) = +/- 2sqrt(2)

(2sqrt(2) , 0)

(-2sqrt(2) , 0)

Foci: (c , 0)

c² = a² - b²

c² = 8 - 3 = 5

c = +/- sqrt(5)

Answer 2

The center is at (0,0)

The vertices are at ( ( ±2 sqrt(2),0)

foci are ( ±sqrt(5),0)

Explanation:

3x^2 + 8y^2 = 24

Divide each side by 24

3x^2 /24 + 8y^2/24 = 24/24

x^2/8 + y^2 /3 = 1

The general equation of an ellipse is

(x-h)^2/ a^2 + (y-k)^2 / b^2 = 1

a>b (h,k) is the center

the coordinates of the vertices are ( ±a,0)

the coordinates of the foci are ( ±c,0), where ^c2=a^2−b^ 2

The center is at (0,0)

a = sqrt(8) = 2sqrt(2)

The vertices are at ( ( ±2 sqrt(2),0)

c = 8 - 3 =5

foci are ( ±sqrt(5),0)