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Chapter 6: Problem 24

Find the standard form of the equation of the ellipse with the given characteristics. Center: (2,-1)\(;\) vertex: \(\left(2, \frac{1}{2}\right) ;\) minor axis of length 2

Short Answer

Expert verified
The standard form of the equation of the ellipse is 鈥嶾(4/9 (x - 2)^2 + (y + 1)^2 = 1\).

Step by step solution

01

Determine the values of a and b

Since a vertex is (2, 1/2), this indicates the semi-major axis 'a'. Since the center is (2, -1) and one vertex is at (2, 1/2), the distance between them is the absolute difference between their respective y-values (\(1/2\) - (-1)), which gives us the value of 'a' as 鈥嶾(3/2\). The length of the minor axis is given as 2, indicating 2b. Hence, b is 1.
02

Substitute the values into the standard equation of the ellipse

Now, substitute the values of h, k, a and b into the standard equation of an ellipse to get your final equation. Here, h = 2, k = -1, a = 鈥嶾(3/2\), and b = 1. So, substituting in these values gives \((x - 2)^2 / (3/2)^2 + (y - (-1))^2 / 1^2 = 1\). Hence, the standard form of the equation of the ellipse is 鈥嶾(4/9 (x - 2)^2 + (y + 1)^2 = 1\).

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Most popular questions from this chapter

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