The problem states that:

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

Solution:

The general equation of a circle is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius. Here, $r = \frac{1}{2}\sqrt{10}$.

Step 1: Center lies on $y = 3x - 7$

Since the center $(h, k)$ lies on the line $y = 3x - 7$, we can write: $k = 3h - 7$

Step 2: Circle passes through $(1, 1)$

The point $(1, 1)$ satisfies the circle's equation: $(1 - h)^2 + (1 - k)^2 = \left(\frac{1}{2}\sqrt{10}\right)^2$ Simplify the radius term: $\left(\frac{1}{2}\sqrt{10}\right)^2 = \frac{10}{4} = \frac{5}{2}$ Thus: $(1 - h)^2 + (1 - k)^2 = \frac{5}{2}$

Step 3: Substitute $k = 3h - 7$

Substitute $k = 3h - 7$ into the equation: $(1 - h)^2 + \left(1 - (3h - 7)\right)^2 = \frac{5}{2}$ Simplify $1 - (3h - 7)$: $1 - (3h - 7) = 1 - 3h + 7 = 8 - 3h$ So the equation becomes: $(1 - h)^2 + (8 - 3h)^2 = \frac{5}{2}$

Step 4: Expand and simplify

Expand $(1 - h)^2$: $(1 - h)^2 = 1 - 2h + h^2$ Expand $(8 - 3h)^2$: $(8 - 3h)^2 = 64 - 48h + 9h^2$ Substitute these into the equation: $1 - 2h + h^2 + 64 - 48h + 9h^2 = \frac{5}{2}$ Simplify: $10h^2 - 50h + 65 = \frac{5}{2}$ Multiply through by 2 to eliminate the fraction: $20h^2 - 100h + 130 = 5$ Simplify further: $20h^2 - 100h + 125 = 0$ Divide through by 5: $4h^2 - 20h + 25 = 0$

Step 5: Solve for $h$

This is a quadratic equation: $4h^2 - 20h + 25 = 0$ Factorize or use the quadratic formula: $h = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(4)(25)}}{2(4)}$ $h = \frac{20 \pm \sqrt{400 - 400}}{8}$ $h = \frac{20}{8} = 2.5$

Step 6: Find $k$

Substitute $h = 2.5$ into $k = 3h - 7$: $k = 3(2.5) - 7 = 7.5 - 7 = 0.5$

Step 7: Write the equation of the circle

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

Do you want further explanation on any of the steps?

Related Questions:

1. What is the formula for a circle given its center and radius?
2. How do you find the intersection of a line and a circle?
3. How can you determine if a point lies inside, on, or outside a circle?
4. What are some real-world applications of circles in geometry?
5. How can the quadratic formula be used to find roots of equations?

Tip:

Always substitute intermediate results back into constraints to ensure they satisfy all conditions.