Final answer:
The question is about forming the equation of an ellipse with the major axis endpoints and the distance to a focus. The semi-major axis is 5, and using the distance to a focus, c = 4, the semi-minor axis b is calculated to be 3. The equation of the ellipse is x²/25 + y²/9 = 1.
Step-by-step explanation:
The student's question is about finding the equation of an ellipse with given endpoints of the major axis and the distance from the center to a focus (c). The major axis endpoints at (-5, 0) and (5, 0) indicate that the major axis is horizontal and the center of the ellipse is at the origin (0, 0). The distance from the center of the ellipse to a focus is given by c = 4. We can use this information along with the fact that the distance c is related to the semi-major axis a and semi-minor axis b of an ellipse through the equation c² = a² - b². The length of the semi-major axis a is half the distance between the endpoints of the major axis, which is 5 in this case, so a = 5. We can then find the semi-minor axis b using the given value of c.
Therefore, the equation of the ellipse can be found using the formula:
²/a² + y²/b² = 1
First, we calculate b using c² = a² - b²:
16 = 25 - b²
b² = 25 - 16
b² = 9
Thus, b = 3.
The final equation of the ellipse is:
x²/25 + y²/9 = 1