To solve the problem of finding the number of points of intersection between the two graphs, let's analyze the equations provided:

1. Equation 1: $(x - 4)^2 + (y - 3)^2 = 4$
   This represents a circle with center $(4, 3)$ and radius $2$ (since $r^2 = 4$).

2. Equation 2: $\frac{(x - 4)^2}{4} + \frac{(y - 10)^2}{16} = 1$
   This represents an ellipse centered at $(4, 10)$, with a semi-major axis of $4$ (along the $y$-axis) and a semi-minor axis of $2$ (along the $x$-axis).

Step-by-Step Solution:

1. Location of the circle and ellipse centers:

   • The circle's center is at $(4, 3)$.
   • The ellipse's center is at $(4, 10)$.
   • The vertical distance between the centers is $10 - 3 = 7$.

2. Range of the shapes:

   • The circle extends vertically from $y = 3 - 2 = 1$ to $y = 3 + 2 = 5$.
   • The ellipse extends vertically from $y = 10 - 4 = 6$ to $y = 10 + 4 = 14$.

   Clearly, the circle and the ellipse do not overlap vertically. The circle's highest point is at $y = 5$, and the ellipse's lowest point is at $y = 6$.

Conclusion:

The circle and the ellipse do not intersect, so the number of intersection points is 0.

Answer: F. 0

Let me know if you'd like further clarification on this solution.

Here are 5 related questions to expand your understanding:

1. How do you determine if two curves intersect in general?
2. What are the differences between the equations of circles and ellipses?
3. How would the solution change if the circle had a larger radius?
4. What happens if the centers of the circle and ellipse are the same?
5. How can we graphically verify the solution using software or tools?

Tip: Always check for vertical or horizontal overlap when determining intersections of geometric shapes.