Understanding the Geometry of the Rectangle:
\(C\) and \(D\) are on the x-axis with coordinates \((-2, 0)\) and \((2, 0)\). The length of side \(CD\) (the base of the rectangle) is:
\[ \text{Length of } CD = |2 - (-2)| = 4 \]
Determine the Height of the Rectangle: The area of a rectangle is given by:
\[ \text{Area} = \text{Base} \times \text{Height} \]
Substituting the known values:
\[ 24 = 4 \times \text{Height} \]
\[ \text{Height} = \frac{24}{4} = 6 \]
Finding the Line \(AB\):
Since the height is perpendicular to \(CD\) and the rectangle is symmetric about the x-axis, the lines \(AB\) and \(CD\) are parallel. The line \(AB\) lies at a height of \(\frac{6}{2} = 3\) units above the x-axis. The equation of \(AB\) is:
\[ y = 3 \]
Thus, the best description of the equation of \(AB\) is \(y = 3\).
The center of a circle $ C $ is at the center of the ellipse $ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, where $ a>b $. Let $ C $ pass through the foci $ F_1 $ and $ F_2 $ of $ E $ such that the circle $ C $ and the ellipse $ E $ intersect at four points. Let $ P $ be one of these four points. If the area of the triangle $ PF_1F_2 $ is 30 and the length of the major axis of $ E $ is 17, then the distance between the foci of $ E $ is:
A | B | C | D | Average |
---|---|---|---|---|
3 | 4 | 4 | ? | 4 |
3 | ? | 5 | ? | 4 |
? | 3 | 3 | ? | 4 |
? | ? | ? | ? | 4.25 |
4 | 4 | 4 | 4.25 |