Find an equation for the ellipse. Graph the equation.
Center at left parenthesis 4 comma 3 right parenthesis; focus at left parenthesis 4 comma 9 right parenthesis; contains the point left parenthesis 8 comma 3 right parenthesis
The general equation of an ellipse with center at (h, k), focus at (h, k + c), and containing a point (p, q) can be written as:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Using the provided information:
Center: (4, 3)
Focus: (4, 9)
Point on ellipse: (8, 3)
We first need to calculate the distance between the center and the focus to find c:
c = distance between (4, 3) and (4, 9) = 6
Next, we need to calculate a by using the distance between the center and the given point (8, 3):
a = distance between (4, 3) and (8, 3) = 4
Now we can find b using the relationship between a, b, and c:
c^2 = a^2 - b^2
6^2 = 4^2 - b^2
36 = 16 - b^2
b^2 = 16
b = 4
Substitute the values of a, b, h, k into the general equation:
(x - 4)^2 / 16 + (y - 3)^2 / 16 = 1
The equation of the ellipse is:
(x - 4)^2 / 16 + (y - 3)^2 / 16 = 1
Graph:
```python
import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(0, 8, 400)
y = 3 + 4*np.sqrt(1 - (x-4)**2/16)
plt.plot(x, y, label='Ellipse')
plt.scatter([4, 4, 8], [3, 9, 3], color='red', label='Center, Focus, Point')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Ellipse')
plt.legend()
plt.axis('equal')
plt.show()
```