1. **State the problem:** We have a circle $k$ with center at $(-1, 3)$ and a point on the circle at $(2, 7)$. We need to find: (i) The radius of the circle. (ii) Two other points on the circle with integer coordinates. 2. **Formula for radius:** The radius $r$ of a circle is the distance between its center $(x_1, y_1)$ and any point $(x_2, y_2)$ on the circle: $$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the radius:** Center: $(-1, 3)$, Point on circle: $(2, 7)$ $$r = \sqrt{(2 - (-1))^2 + (7 - 3)^2} = \sqrt{(2 + 1)^2 + 4^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 4. **Equation of the circle:** $$(x + 1)^2 + (y - 3)^2 = 5^2 = 25$$ 5. **Find two other points with integer coordinates on the circle:** We look for integer pairs $(a, b)$ satisfying: $$(a + 1)^2 + (b - 3)^2 = 25$$ Possible integer values for $(a + 1, b - 3)$ are pairs whose squares sum to 25: $$(\pm 3, \pm 4), (\pm 4, \pm 3), (\pm 5, 0), (0, \pm 5)$$ 6. **Calculate points:** - For $(3, 4)$: $a + 1 = 3 \Rightarrow a = 2$, $b - 3 = 4 \Rightarrow b = 7$ (given point) - For $(-3, 4)$: $a + 1 = -3 \Rightarrow a = -4$, $b - 3 = 4 \Rightarrow b = 7$ - For $(3, -4)$: $a + 1 = 3 \Rightarrow a = 2$, $b - 3 = -4 \Rightarrow b = -1$ - For $(-3, -4)$: $a + 1 = -3 \Rightarrow a = -4$, $b - 3 = -4 \Rightarrow b = -1$ - For $(4, 3)$: $a + 1 = 4 \Rightarrow a = 3$, $b - 3 = 3 \Rightarrow b = 6$ - For $(-4, 3)$: $a + 1 = -4 \Rightarrow a = -5$, $b - 3 = 3 \Rightarrow b = 6$ - For $(4, -3)$: $a + 1 = 4 \Rightarrow a = 3$, $b - 3 = -3 \Rightarrow b = 0$ - For $(-4, -3)$: $a + 1 = -4 \Rightarrow a = -5$, $b - 3 = -3 \Rightarrow b = 0$ - For $(5, 0)$: $a + 1 = 5 \Rightarrow a = 4$, $b - 3 = 0 \Rightarrow b = 3$ - For $(-5, 0)$: $a + 1 = -5 \Rightarrow a = -6$, $b - 3 = 0 \Rightarrow b = 3$ - For $(0, 5)$: $a + 1 = 0 \Rightarrow a = -1$, $b - 3 = 5 \Rightarrow b = 8$ - For $(0, -5)$: $a + 1 = 0 \Rightarrow a = -1$, $b - 3 = -5 \Rightarrow b = -2$ 7. **Choose two points different from $(2, 7)$:** Examples: $(-4, 7)$ and $(3, 6)$ **Final answers:** (i) Radius $r = 5$ (ii) Two other points on the circle: $(-4, 7)$ and $(3, 6)$