An ellipse is a stretched circle, and its standard form equation helps us easily identify its key features. The standard form equation is either $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ or $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$, where $(h, k)$ is the center, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis. The major axis is always longer than the minor axis. If $a^2$ is under the $x$ term, the ellipse is horizontal. If $a^2$ is under the $y$ term, the ellipse is vertical.
Think of it like this: $(h,k)$ tells you where the ellipse is centered, and $a$ and $b$ tell you how far it stretches horizontally and vertically. Understanding this form unlocks a world of ellipse-related problems!
Match each term with its definition:
| Term | Definition |
|---|---|
| 1. Center | A. The longer axis of the ellipse. |
| 2. Major Axis | B. The point at the middle of the ellipse. |
| 3. Minor Axis | C. The shorter axis of the ellipse. |
| 4. Semi-major Axis | D. Half the length of the major axis. |
| 5. Semi-minor Axis | E. Half the length of the minor axis. |
Match the numbers to the correct letters! For example: 1 - B
Complete the paragraph with the correct words:
The standard form of an ellipse's equation allows us to identify the __________, semi-major axis, and __________ . If the larger denominator is under the $x^2$ term, the ellipse is __________. The center is represented by the coordinates (__, __).
Word Bank: Vertical, Horizontal, (h, k), Center, Semi-minor Axis
Imagine you have an ellipse with the equation $\frac{(x-2)^2}{9} + \frac{(y+1)^2}{4} = 1$. Explain how changing the '9' to a '16' would affect the ellipse's shape and orientation.
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