Find the center, vertices, and foci of the ellipse with equation 4x2 + 9y2 = 36.

Question

Find the center, vertices, and foci of the ellipse with equation 4x2 + 9y2 = 36.

Center: (0, 0); Vertices: (-3, 0), (3, 0); Foci: Ordered pair negative square root 5 comma 0 and ordered pair square root 5 comma 0
Center: (0, 0); Vertices: (-9, 0), (9, 0); Foci: Ordered pair negative square root 65 comma 0 and ordered pair square root 65 comma 0
Center: (0, 0); Vertices: (0, -3), (0, -3); Foci: Ordered pair 0 comma negative square root 5 and ordered pair 0 comma square root 5
Center: (0, 0); Vertices: (0, -9), (0, 9); Foci: Ordered pair 0 comma negative square root 65 and ordered pair 0 comma square root 65

Answer 1

The standard form of an ellipse's equation is:

 

(x - h)²/(a²) + (y - k)²/(b²) = 1,

 

where (h, k) is the ellipse's center, 2a is the length of the ellipse's horizontal axis, and 2b is the length of the ellipse's vertical axis. With this information, we can find the ellipse's vertices and foci.

 

 

 

So, let's put your equation into standard form by dividing both sides by 36:

 

(1/36)(4x² + 9y²) = 1

(4x²)/36 + (9y²)/36 = 1

x²/9 + y²/4 = 1

 

This means that (h, k) = (0, 0), the ellipse's center.

 

Also, a² = 9, so a = 3, half the length of the ellipse's horizontal axis. Finally, b² = 4, so b = 2, half the length of the ellipse's vertical axis. We call a the semi-major axis and b the semi-minor axis, since a > b.

 

 

 

Let's use this information to find the vertices of the ellipse. The vertices are located along the ellipse's major axis, which extends a length of a = 3 units left and right of the center. So, the vertices are (0+3, 0) and (0-3, 0), or (3, 0) and (-3, 0).

 

 

 

That already tells you that answer choice (A) is correct, but let's find the foci anyway, just in case. The foci are found by calculating c, the distance from focus to center:

 

c² = a² - b²

c² = (3)² - (2)²

c² = 9 - 4

c² = 5

c = √5

 

That means the foci are at (0+√5, 0) and (0-√5, 0), or (√5, 0) and (-√5, 0). That checks out!