The solution to the system of equations gives us the center \((h, k)\) and radius \(r_1\) of the required circle. The results are:

- The center of the circle is \((-1, 3)\).
- The radius of the circle is \(\sqrt{5}\).

### Therefore, the equation of the circle that passes through the point \((1,4)\) and is tangent to the given circle at the point \((-2,1)\) is:

\[
(x + 1)^2 + (y - 3)^2 = 5
\]

Would you like any more details or have any other questions?

Here are 5 related questions you might consider:

1. How can you verify if a point lies on a given circle?
2. What are the geometric implications of two circles being tangent?
3. How would the solution change if the circles were externally tangent?
4. How can you determine the tangent points of two intersecting circles?
5. How does changing the radius of a circle affect its equation?

**Tip:** When working with systems of equations, especially with geometric problems, always check that the solutions make sense in the context of the problem by substituting them back into the original equations.

Discover the solution to a system of equations involving the center and radius of a circle. This problem explores geometric concepts, including circle equations and tangency. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation