1. **Problem statement:** Find the equation of the circle that passes through the origin and cuts off intercepts 3 and 4 from the positive parts of the x-axis and y-axis respectively. 2. **Understanding the problem:** The circle cuts intercepts 3 and 4 on the positive x and y axes, so it passes through points $(3,0)$ and $(0,4)$. 3. **General form of circle equation:** The equation of a circle is $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h,k)$ is the center and $r$ is the radius. 4. **Using the intercepts:** Since the circle passes through $(3,0)$ and $(0,4)$, these points satisfy the equation: $$ (3 - h)^2 + (0 - k)^2 = r^2 $$ $$ (0 - h)^2 + (4 - k)^2 = r^2 $$ 5. **Also, the circle passes through the origin $(0,0)$:** $$ (0 - h)^2 + (0 - k)^2 = r^2 $$ 6. **Set up equations:** From the three points, we have: $$ (3 - h)^2 + k^2 = r^2 $$ $$ h^2 + (4 - k)^2 = r^2 $$ $$ h^2 + k^2 = r^2 $$ 7. **Subtract the third equation from the first two:** $$ (3 - h)^2 + k^2 - (h^2 + k^2) = 0 \Rightarrow (3 - h)^2 - h^2 = 0 $$ $$ h^2 + (4 - k)^2 - (h^2 + k^2) = 0 \Rightarrow (4 - k)^2 - k^2 = 0 $$ 8. **Simplify:** $$ (3 - h)^2 - h^2 = 0 \Rightarrow 9 - 6h + h^2 - h^2 = 0 \Rightarrow 9 - 6h = 0 \Rightarrow h = \frac{9}{6} = 1.5 $$ $$ (4 - k)^2 - k^2 = 0 \Rightarrow 16 - 8k + k^2 - k^2 = 0 \Rightarrow 16 - 8k = 0 \Rightarrow k = \frac{16}{8} = 2 $$ 9. **Find radius $r$ using $h$ and $k$ in the third equation:** $$ r^2 = h^2 + k^2 = (1.5)^2 + 2^2 = 2.25 + 4 = 6.25 $$ 10. **Final equation:** $$ (x - 1.5)^2 + (y - 2)^2 = 6.25 $$ This is the equation of the circle passing through the origin and cutting intercepts 3 and 4 on the positive axes.