Question 1185775
Here's how to solve this problem:

**Understanding the Geometry**

A whispering gallery is shaped like an ellipse. The foci are the points where whispers can be heard clearly.  The key property we'll use is that the *sum* of the distances from any point on the ellipse to the two foci is constant.

**a. Finding the Highest Point of the Ceiling**

1.  **Distance to Foci:** Yohan is 5 feet from the nearest wall (which is also a vertex of the ellipse), and Jesx is 9 feet away from Yohan. This means the distance between the foci (2c) is 9 feet.  Therefore, c = 9/2 = 4.5 feet.

2.  **Distance from Focus to Vertex:** Yohan is 5 feet from the nearest wall, which is a vertex. The distance from a focus to the nearest vertex is *a - c*, where 'a' is the semi-major axis of the ellipse. So, a - c = 5.

3.  **Solving for 'a':**  a - 4.5 = 5  => a = 9.5 feet.

4.  **Center of the Ellipse:** The center of the ellipse is halfway between the foci. Since Yohan is 5 feet from the nearest wall (vertex), and the distance between the foci is 9 feet, the distance from Yohan to the center is (9/2) + 5 - 9.5 = 5 feet.

5.  **Semi-minor Axis (b):** We can find 'b' using the relationship in an ellipse:  c² = a² - b².
    4.5² = 9.5² - b²
    b² = 9.5² - 4.5²
    b² = 90 - 20.25
    b²= 70.75
    b = √70.75 ≈ 8.41 feet

6.  **Height of the Ceiling:** The highest point of the ceiling is the distance from the center of the ellipse to the top, which is equal to the semi-minor axis *b*. Therefore, the highest point of the ceiling from the ground is approximately 8.41 feet.

**b. Height of the Ceiling Where Yohan and Jesx Stand**

1.  **Horizontal Distance:** Yohan is at one focus. The foci are on the horizontal axis.  So, the horizontal distance from the center of the ellipse to where Yohan stands is *c* = 4.5 feet.

2.  **Equation of the Ellipse:** If we place the center of the ellipse at the origin (0,0), the equation of the ellipse is:

    (x²/a²) + (y²/b²) = 1
    (x²/9.5²) + (y²/70.75) = 1

3.  **Substitute x = 4.5 and solve for y:**  We want to find the height *y* when *x* = 4.5 (where Yohan stands).
    (4.5²/9.5²) + (y²/70.75) = 1
    0.225 + (y²/70.75) = 1
    y²/70.75 = 0.775
    y² = 54.84
    y = √54.84 ≈ 7.4 feet

Therefore, the height of the ceiling where Yohan and Jesx stand is approximately 7.4 feet.

**Final Answers:**

*   **a.** The highest point of the ceiling from the ground is approximately 8.41 feet.
*   **b.** The height of the ceiling from the point where Yohan and Jesx stand is approximately 7.4 feet.