Given the equation x 2 4 + y 2 9 = 1 \frac{x^2}{4}+\frac{y^2}{9}=1 4 x 2 + 9 y 2 = 1 , sketch a graph of the ellipse.

let's see if we can solve this problem. In this problem, we are given the equation x squared over four plus y squared over nine is equal to one, and we're asked to sketch a graph of the ellipse that is formed by this equation. Now to start things off, I noticed that the larger number in the denominator is nine, between nine and four. And nine will be associated with the semi major axis a. And because this a is underneath the y term, I know that this ellipse that we're going to get is going to be a vertical ellipse stretched on the y axis. I also see that we have an x squared over something plus y squared over another thing, which tells me we our ellipse is going to be centered at the origin. So these are the details that I can see from this equation. Now nine is going to be a squared, meaning that four over here is going to be b squared. So what I can do is set a squared equal to nine and solve for a. If I take the square root on both sides of this equation, that will give me that a is equal to the square root of nine, which is three. And if I take b squared instead of equal to four, I can take the square root on both sides of this equation and get that b is equal to the square root of four, which is two. So we now have our a value of three, which is the semi major axis, and then our b value of two, which is the semi minor axis. And at this point, we can graph our ellipse. So if I go over here to the graph, we're going to be centered at the origin. And here we have the semi major axis of three. Recall since we're stretched on the y axis by this equation up here, we're going to go one, two, three units up, and one, two, three units down for the semi major axis. Now for the semi minor axis, we're going to go one, two units to the left and one, two units to the right. So our ellipse is going to look something like this. And this is how you can graph an ellipse centered at the origin based on the equation. So hope you found this helpful, and let's move forward.

16. Parametric Equations & Polar Coordinates

Conic Sections

Calculus

16. Parametric Equations & Polar Coordinates

Conic Sections

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Multiple Choice

Given the equation $\frac{x^2}{4}+\frac{y^2}{9}=1$ , sketch a graph of the ellipse.

Option A

option B

option C

option d

Verified step by step guidance

Step 1: Recognize the given equation $ \frac{x^2}{4} + \frac{y^2}{9} = 1 $. This is the standard form of an ellipse centered at the origin $(0, 0)$.

Step 2: Identify the denominators under $x^2$ and $y^2$. The denominator under $x^2$ is 4, and the denominator under $y^2$ is 9. These values represent the squares of the semi-axes lengths.

Step 3: Determine the semi-major and semi-minor axes. Since $9 > 4$, the semi-major axis is along the $y$-axis with length $\sqrt{9} = 3$, and the semi-minor axis is along the $x$-axis with length $\sqrt{4} = 2$.

Step 4: Sketch the ellipse. Draw the ellipse centered at $(0, 0)$, extending 3 units vertically (up and down along the $y$-axis) and 2 units horizontally (left and right along the $x$-axis).

Step 5: Compare the provided graphs. The correct graph should show an ellipse elongated along the $y$-axis with a vertical semi-major axis of 3 and a horizontal semi-minor axis of 2.