Key Concepts and Formulas

• Equation of a Circle: A circle with center $(h,k)$ and radius $r$ has the equation $(x-h)^2 + (y-k)^2 = r^2$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Conditions for Tangency between Two Circles: Let two circles have centers $C_1, C_2$ and radii $r_1, r_2$ respectively. The distance between their centers is $d = C_1C_2$.
  • External Tangency: If the circles touch externally, then $d = r_1 + r_2$.
  • Internal Tangency: If one circle touches the other internally, then $d = |r_1 - r_2|$.

2. Step-by-Step Solution

Step 1: Define the first circle and use the given points to establish relationships between h, k, and r.

We are given that the circle passes through the points $(0,0)$ and $(1,0)$. Let the center of the circle be $(h,k)$ and its radius be $r$. The equation of the circle is $(x-h)^2 + (y-k)^2 = r^2$.

• Using the point $(0,0)$: Since the circle passes through $(0,0)$, we substitute these coordinates into the circle's equation: $(0-h)^2 + (0-k)^2 = r^2$ $h^2 + k^2 = r^2$ This is a key relationship. It tells us that the square of the radius is equal to the sum of the squares of the coordinates of the center.

• Using the point $(1,0)$: Since the circle passes through $(1,0)$, we substitute these coordinates into the circle's equation: $(1-h)^2 + (0-k)^2 = r^2$ $(1-h)^2 + k^2 = r^2$

• Finding the value of $h$: We have two expressions for $r^2$. Equating them will allow us to find a relationship between $h$ and $k$: $h^2 + k^2 = (1-h)^2 + k^2$ Subtract $k^2$ from both sides: $h^2 = (1-h)^2$ Expand the right side: $h^2 = 1 - 2h + h^2$ Subtract $h^2$ from both sides: $0 = 1 - 2h$ $2h = 1$ $h = \frac{1}{2}$ This result is significant: the x-coordinate of the center is $1/2$.

• Expressing $r^2$ in terms of $k$: Substitute $h = 1/2$ back into the equation $r^2 = h^2+k^2$: $r^2 = \left(\frac{1}{2}\right)^2 + k^2$ $r^2 = \frac{1}{4} + k^2$ Thus, the radius of our circle is $r = \sqrt{\frac{1}{4} + k^2}$.

Step 2: Analyze the second circle and identify its center and radius.

The second circle is given by the equation $x^2+y^2=9$.

• Its center, $C_2$, is $(0,0)$.
• Its radius, $R_2$, is $\sqrt{9} = 3$.

Step 3: Apply the tangency condition between the two circles.

Let our circle be $C_1$ with center $(h,k) = \left(\frac{1}{2}, k\right)$ and radius $r = \sqrt{\frac{1}{4}+k^2}$. Let the given circle be $C_2$ with center $(0,0)$ and radius $R_2 = 3$.

The distance between the centers $C_1$ and $C_2$ is: $d = \sqrt{\left(\frac{1}{2}-0\right)^2 + (k-0)^2} = \sqrt{\frac{1}{4}+k^2}$ Notice that this distance $d$ is exactly equal to the radius $r$ of our first circle. So, $d=r$.

Now we apply the tangency conditions:

• Case A: External Tangency If the circles touch externally, then the distance between their centers equals the sum of their radii: $d = r + R_2$ Substitute $d=r$ and $R_2=3$: $r = r + 3$ $0 = 3$ This is a contradiction, meaning external tangency is not possible.

• Case B: Internal Tangency If the circles touch internally, then the distance between their centers equals the absolute difference of their radii: $d = |r - R_2|$ Substitute $d=r$ and $R_2=3$: $r = |r - 3|$ This equation implies two possibilities:

  1. $r = r - 3$ (which leads to $0 = -3$, a contradiction, so this is impossible)
  2. $r = -(r - 3)$ $r = -r + 3$ $2r = 3$ $r = \frac{3}{2}$ This gives us a valid radius for our circle.

Now we know the radius of our circle is $r = \frac{3}{2}$. From Step 1, we established that $r^2 = \frac{1}{4} + k^2$. So, we can write: $\left(\frac{3}{2}\right)^2 = \frac{1}{4} + k^2$ $\frac{9}{4} = \frac{1}{4} + k^2$ $k^2 = \frac{9}{4} - \frac{1}{4}$ $k^2 = \frac{8}{4} = 2$

Since $h = \frac{1}{2}$, we have $h^2 = \frac{1}{4}$. Thus, $h^2 + k^2 = \frac{1}{4} + 2 = \frac{1}{4} + \frac{8}{4} = \frac{9}{4}$

Step 4: Calculate the final expression.

The problem asks for the value of $4(h^2+k^2)$. Using the value we just found for $h^2+k^2$: $4\left(h^2+k^2\right) = 4 \times \frac{9}{4}$ $4\left(h^2+k^2\right) = 9$

3. Common Mistakes & Tips

• Incorrect Tangency Condition: Forgetting the absolute value in the internal tangency condition $d = |r_1 - r_2|$ can lead to incorrect solutions.
• Algebra Errors: Be extra careful with algebraic manipulations, especially when expanding squares or solving equations with square roots.
• Missing Cases: Always consider both external and internal tangency unless the problem statement explicitly restricts it.

4. Summary

We determined the center and radius of a circle passing through $(0,0)$ and $(1,0)$ and touching the circle $x^2+y^2=9$. By using the equation of a circle, distance formula, and tangency conditions, we derived that $h^2+k^2 = \frac{9}{4}$. Therefore, $4(h^2+k^2) = 9$.

5. Final Answer

The final answer is \boxed{9}.