1. **State the problem:** We are given the line equation $y = -\frac{4}{3}x + 4$ and need to: (i) Show that the coordinates of $C$ are $(3,0)$. (ii) Find the equation of circle $C_1$ with center $C$ passing through points $A(0,4)$ and $B(8,0)$. (iii) Given $GF$ is the diameter of circle $C_2$ passing through $C$, $F$, and $G$, calculate the radius of $C_2$. 2. **Show that $C = (3,0)$ lies on the line:** Substitute $x=3$ into the line equation: $$y = -\frac{4}{3} \times 3 + 4 = -4 + 4 = 0$$ So, $C = (3,0)$ satisfies the line equation. 3. **Find the equation of circle $C_1$ with center $C(3,0)$ passing through $A(0,4)$:** The radius $r$ is the distance from $C$ to $A$: $$r = \sqrt{(0-3)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 4. **Write the equation of circle $C_1$:** The general form is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ where center is $(h,k) = (3,0)$ and $r=5$: $$ (x - 3)^2 + (y - 0)^2 = 25 $$ 5. **Calculate the radius of circle $C_2$ with diameter $GF$:** Since $GF$ is the diameter, radius $r_2 = \frac{1}{2} \times GF$. 6. **Find coordinates of $F$ and $G$:** Points $F$ and $G$ lie on circle $C_1$ and on the normal line $y = -\frac{4}{3}x + 4$. Since $C$ is center and $F$ lies on the circle and line, find $F$ by solving: $$ (x - 3)^2 + \left(-\frac{4}{3}x + 4\right)^2 = 25 $$ Expand and simplify: $$ (x - 3)^2 + \left(-\frac{4}{3}x + 4\right)^2 = 25 $$ $$ (x - 3)^2 + \left(-\frac{4}{3}x + 4\right)^2 = 25 $$ Calculate each term: $$ (x - 3)^2 = x^2 - 6x + 9 $$ $$ \left(-\frac{4}{3}x + 4\right)^2 = \left(-\frac{4}{3}x\right)^2 - 2 \times \frac{4}{3}x \times 4 + 4^2 = \frac{16}{9}x^2 - \frac{32}{3}x + 16 $$ Sum: $$ x^2 - 6x + 9 + \frac{16}{9}x^2 - \frac{32}{3}x + 16 = 25 $$ Multiply all terms by 9 to clear denominators: $$ 9x^2 - 54x + 81 + 16x^2 - 96x + 144 = 225 $$ Combine like terms: $$ 25x^2 - 150x + 225 = 225 $$ Subtract 225 from both sides: $$ 25x^2 - 150x + 225 - 225 = 0 $$ $$ 25x^2 - 150x = 0 $$ Divide both sides by 25: $$ \cancel{25}x^2 - \cancel{25}6x = 0 $$ $$ x^2 - 6x = 0 $$ Factor: $$ x(x - 6) = 0 $$ So, $x=0$ or $x=6$. 7. **Find corresponding $y$ values:** For $x=0$: $$ y = -\frac{4}{3} \times 0 + 4 = 4 $$ So $F = (0,4)$. For $x=6$: $$ y = -\frac{4}{3} \times 6 + 4 = -8 + 4 = -4 $$ So $G = (6,-4)$. 8. **Calculate length $GF$:** $$ GF = \sqrt{(6-0)^2 + (-4-4)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $$ 9. **Calculate radius of circle $C_2$:** $$ r_2 = \frac{GF}{2} = \frac{10}{2} = 5 $$ **Final answers:** (i) $C = (3,0)$ lies on the line. (ii) Equation of circle $C_1$: $$ (x - 3)^2 + y^2 = 25 $$ (iii) Radius of circle $C_2$ is $5$.