Let's go through each problem in the image you've shared:

Question 1:

"The line $y = 2x - 8$ meets the coordinate axes at points $A$ and $B$. The line $AB$ is a diameter of the circle. Find the equation of the circle."

• Step 1: Find intercepts (A and B)

  • When the line meets the x-axis, $y = 0$: $0 = 2x - 8 \implies x = 4$ So, point $A = (4, 0)$.
  • When the line meets the y-axis, $x = 0$: $y = 2(0) - 8 = -8$ So, point $B = (0, -8)$.

• Step 2: Find the center and radius of the circle

  • The center of the circle is the midpoint of the diameter $AB$: $\text{Midpoint of } A(4, 0) \text{ and } B(0, -8) = \left( \frac{4 + 0}{2}, \frac{0 + (-8)}{2} \right) = (2, -4)$
  • The radius is half the distance of $AB$: $\text{Distance } AB = \sqrt{(4 - 0)^2 + (0 - (-8))^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$ So, the radius $r = \frac{4\sqrt{5}}{2} = 2\sqrt{5}$.

• Step 3: Equation of the circle 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.

  • Center $(2, -4)$
  • Radius $r = 2\sqrt{5}$, so $r^2 = 20$.

  The equation of the circle is: $(x - 2)^2 + (y + 4)^2 = 20$

Question 2:

"The circle center $(8, 10)$ meets the x-axis at $(4, 0)$ and $(a, 0)$."

• Part (a): Find the radius of the circle.

  • The radius is the distance from the center $(8, 10)$ to any point on the circle, such as $(4, 0)$. $r = \sqrt{(8 - 4)^2 + (10 - 0)^2} = \sqrt{16 + 100} = \sqrt{116}$ So, the radius is $\sqrt{116}$, which simplifies to $2\sqrt{29}$.

• Part (b): Find the value of $a$.

  • The radius must also be the distance from the center $(8, 10)$ to $(a, 0)$. We know the radius is $\sqrt{116}$, so: $\sqrt{(8 - a)^2 + (10 - 0)^2} = \sqrt{116}$ Squaring both sides: $(8 - a)^2 + 100 = 116 \implies (8 - a)^2 = 16 \implies 8 - a = \pm 4$ Solving this: $a = 8 - 4 = 4 \quad \text{or} \quad a = 8 + 4 = 12$ Therefore, $a = 12$ (since we already have $a = 4$ as one solution).

Would you like further details on any of these solutions? Here are some questions to expand on this topic:

1. How can we derive the equation of a line from two points?
2. What other ways can we calculate the radius of a circle from its diameter?
3. Why does a circle's equation take the form $(x - h)^2 + (y - k)^2 = r^2$?
4. Can you generalize how to find the intercepts of a line for any linear equation $y = mx + c$?
5. How do you verify if a given point lies on a circle?

Tip: Always check the units of radius and other distances in geometry problems to ensure they are consistent when plugging into formulas.