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 4, plus y squared over 9 is equal to 1, 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 9, between 9 and 4, And 9 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 9 is going to be a squared, meaning that 4 over here is going to be b squared. So what I can do is set a squared equal to 9 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 9, which is 3. And if I take b squared instead of equal to 4, I can take the square root on both sides of this equation, and get that b is equal to the square root of 4, which is 2. So we now have our a value of 3, which is the semi major axis, and then our b value of 2, 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 3. Recall since we're stretched on the y axis by this equation up here, we're going to go 1, 2, 3 units up, and 1, 2, 3 units down for the semi major axis. Now for the semi minor axis, we're going to go 1, 2 units to the left and 1, 2 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.

19. Conic Sections

Ellipses: Standard Form

Precalculus

19. Conic Sections

Ellipses: Standard Form

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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

Identify the standard form of the ellipse equation: $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $. In this case, the equation is $ \frac{x^2}{4} + \frac{y^2}{9} = 1 $.

Determine the values of $ a^2 $ and $ b^2 $ from the equation. Here, $ a^2 = 4 $ and $ b^2 = 9 $.

Calculate the values of $ a $ and $ b $ by taking the square root of $ a^2 $ and $ b^2 $. Thus, $ a = 2 $ and $ b = 3 $.

Since $ b > a $, the major axis is vertical. The ellipse is taller than it is wide.

Sketch the ellipse centered at the origin (0,0) with a vertical major axis of length 6 (2b) and a horizontal minor axis of length 4 (2a).

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