Let us solve the given problem step by step.

Problem:

A circle with radius $\frac{1}{2}\sqrt{10}$ passes through the point $(1, 1)$, and the center of the circle lies on the straight line $y = 3x - 7$. Find the equation of the circle.

Step-by-Step Solution:

1. General equation of a circle: The equation of a circle is given by: $(x - h)^2 + (y - k)^2 = r^2$ Here, $(h, k)$ is the center, and $r$ is the radius.

   For this problem:

   • Radius $r = \frac{1}{2}\sqrt{10}$,
   • $(h, k)$ is the center, which lies on the line $y = 3x - 7$.

   So, $k = 3h - 7$.

2. Substitute the point $(1, 1)$: Since the circle passes through $(1, 1)$, we substitute $x = 1$, $y = 1$, $r = \frac{1}{2}\sqrt{10}$, and $k = 3h - 7$ into the circle equation: $(1 - h)^2 + (1 - k)^2 = \left(\frac{1}{2}\sqrt{10}\right)^2$ Simplify: $(1 - h)^2 + (1 - (3h - 7))^2 = \frac{10}{4}$ $(1 - h)^2 + (1 - 3h + 7)^2 = \frac{5}{2}$ $(1 - h)^2 + (8 - 3h)^2 = \frac{5}{2}$

3. Expand and simplify: Expand the two squares: $(1 - h)^2 = 1 - 2h + h^2,$ $(8 - 3h)^2 = 64 - 48h + 9h^2.$ Add these: $(1 - 2h + h^2) + (64 - 48h + 9h^2) = \frac{5}{2}.$ Combine terms: $1 + 64 - 2h - 48h + h^2 + 9h^2 = \frac{5}{2}.$ $65 - 50h + 10h^2 = \frac{5}{2}.$

4. Eliminate the fraction: Multiply through by 2 to clear the fraction: $130 - 100h + 20h^2 = 5.$ Rearrange into standard quadratic form: $20h^2 - 100h + 125 = 0.$ Simplify by dividing through by 5: $4h^2 - 20h + 25 = 0.$

5. Solve for $h$: This is a quadratic equation. Solve using the quadratic formula: $h = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(4)(25)}}{2(4)}.$ $h = \frac{20 \pm \sqrt{400 - 400}}{8}.$ $h = \frac{20 \pm 0}{8}.$ $h = \frac{20}{8} = 2.5.$

6. Find $k$: Using $k = 3h - 7$: $k = 3(2.5) - 7 = 7.5 - 7 = 0.5.$

7. Equation of the circle: The center of the circle is $(2.5, 0.5)$, and the radius is $\frac{1}{2}\sqrt{10}$. The equation of the circle is: $(x - 2.5)^2 + (y - 0.5)^2 = \left(\frac{1}{2}\sqrt{10}\right)^2.$ Simplify: $(x - 2.5)^2 + (y - 0.5)^2 = \frac{5}{2}.$

Final Answer:

The equation of the circle is: $(x - 2.5)^2 + (y - 0.5)^2 = \frac{5}{2}.$

Would you like further clarifications or additional details?

Here are 5 related questions for practice:

1. How would you solve this if the radius was unknown?
2. What is the geometric meaning of the quadratic equation derived here?
3. Can a circle have its center off the given line in this problem?
4. Derive the standard form equation of the circle from the final answer.
5. How would you verify the solution graphically?

Tip: Always check your quadratic discriminant to confirm real solutions before proceeding!