1. **State the problem:** We have a circle $C_1$ with diameter endpoints $A(-2,18)$ and $B(14,6)$. We need to find the equation of $C_1$. Then, given another circle $C_2$ centered at the origin $O(0,0)$ that touches $C_1$, find possible equations for $C_2$. 2. **Find the center and radius of $C_1$:** The center of a circle with diameter endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ is the midpoint: $$\text{Center } C = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ Calculate: $$C = \left(\frac{-2+14}{2}, \frac{18+6}{2}\right) = (6, 12)$$ The radius $r$ is half the length of diameter $AB$: $$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(14+2)^2 + (6-18)^2} = \sqrt{16^2 + (-12)^2} = \sqrt{256 + 144} = \sqrt{400} = 20$$ So, $$r = \frac{AB}{2} = 10$$ 3. **Equation of circle $C_1$:** The general equation of a circle with center $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ Substitute $h=6$, $k=12$, $r=10$: $$ (x - 6)^2 + (y - 12)^2 = 100 $$ 4. **Find possible equations for circle $C_2$ centered at origin $O(0,0)$ that touches $C_1$:** Since $C_2$ is centered at origin, its equation is: $$ x^2 + y^2 = R^2 $$ where $R$ is the radius of $C_2$. 5. **Condition for tangency of two circles:** Two circles with centers $C_1(h,k)$ and $C_2(0,0)$ and radii $r$ and $R$ touch if the distance between centers equals sum or difference of radii: $$ d = \sqrt{h^2 + k^2} = r + R \quad \text{(external tangency)} $$ or $$ d = |r - R| \quad \text{(internal tangency)} $$ Calculate distance between centers: $$ d = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5} $$ 6. **Solve for $R$:** - External tangency: $$ 6\sqrt{5} = 10 + R \implies R = 6\sqrt{5} - 10 $$ - Internal tangency: $$ 6\sqrt{5} = |10 - R| $$ This gives two cases: (a) $10 - R = 6\sqrt{5} \implies R = 10 - 6\sqrt{5}$ (b) $R - 10 = 6\sqrt{5} \implies R = 10 + 6\sqrt{5}$ Since radius must be positive, check values: - $6\sqrt{5} \approx 13.416$ - $6\sqrt{5} - 10 \approx 3.416$ (valid) - $10 - 6\sqrt{5} \approx -3.416$ (not valid) - $10 + 6\sqrt{5} \approx 23.416$ (valid) 7. **Possible equations for $C_2$ are:** $$ x^2 + y^2 = (6\sqrt{5} - 10)^2 $$ and $$ x^2 + y^2 = (10 + 6\sqrt{5})^2 $$ **Final answers:** - Equation of $C_1$: $$ (x - 6)^2 + (y - 12)^2 = 100 $$ - Possible equations of $C_2$: $$ x^2 + y^2 = (6\sqrt{5} - 10)^2 $$ $$ x^2 + y^2 = (10 + 6\sqrt{5})^2 $$