Question

Graphed in the standard (x, y) coordinate plane below is an ellipse. The center of the ellipse is (0, 0), and points (-5, 0), (0, 3), (5, 0), (0, -3), A(3, a), and B(3, b) lie on the ellipse. What is the distance, in coordinate units, from A toB ? A. 2.4 B. 3 C. 4 D. 4.8 E. 6

Answer 1

The distance from point A(3, a) to point B(3, b) on the given ellipse is 3 coordinate units.

Step-by-step explanation:

To find the distance between points A and B on the ellipse, we can use the distance formula:
`\(d = √((x_2 - x_1)^2 + (y_2 - y_1)^2)\)`. In this case, the x-coordinates of A and B are both 3, and their y-coordinates are a and b, respectively.

The general equation of an ellipse centered at the origin is given by
`\((x^2)/(a^2) + (y^2)/(b^2) = 1\)`, where a and b are the semi-major and semi-minor axes, respectively. Since the ellipse is centered at (0, 0) and passes through points (-5, 0), (0, 3), (5, 0), and (0, -3), we can deduce that the semi-major axis
`(\(a\))` is 5, and the semi-minor axis
`(\(b\))` is 3.

Now, plug the coordinates of points A and B into the distance formula:

For point A(3, a):
`\(d_A = √((3 - 3)^2 + (a - 0)^2) = |a|\)`

For point B(3, b):
`\(d_B = √((3 - 3)^2 + (b - 0)^2) = |b|\)`

Since both points have the same x-coordinate (3), the distance between them is simply the absolute difference of their y-coordinates, which is
`\(|a - b|\)`. Given that the ellipse passes through (0, 3) and (0, -3), we know
`\(|a - b| = 3\).`

Therefore, the distance from A to B on the ellipse is 3 coordinate units.