1. **Problem statement:** Find the lengths of line segments $OA$ and $OB$ where points $A$ and $B$ are the intersections of the line $y=x$ with the parabolas $y=x^2$ and $y=-\frac{1}{4}x^2$ respectively, besides the origin $O$. 2. **Find coordinates of point $A$:** Set $y=x$ equal to $y=x^2$: $$x = x^2$$ Rearranged: $$x^2 - x = 0$$ Factor: $$x(x - 1) = 0$$ Solutions: $$x=0 \quad \text{or} \quad x=1$$ At $x=0$, point is $O=(0,0)$. At $x=1$, $y=1$, so $A=(1,1)$. 3. **Find coordinates of point $B$:** Set $y=x$ equal to $y=-\frac{1}{4}x^2$: $$x = -\frac{1}{4}x^2$$ Rearranged: $$\frac{1}{4}x^2 + x = 0$$ Multiply both sides by 4 to clear fraction: $$x^2 + 4x = 0$$ Factor: $$x(x + 4) = 0$$ Solutions: $$x=0 \quad \text{or} \quad x=-4$$ At $x=0$, point is $O=(0,0)$. At $x=-4$, $y=-4$, so $B=(-4,-4)$. 4. **Calculate lengths $OA$ and $OB$:** Length formula between points $(x_1,y_1)$ and $(x_2,y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ For $OA$: $$d_{OA} = \sqrt{(1-0)^2 + (1-0)^2} = \sqrt{1 + 1} = \sqrt{2}$$ For $OB$: $$d_{OB} = \sqrt{(-4-0)^2 + (-4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$$ **Final answer:** $$OA = \sqrt{2}, \quad OB = 4\sqrt{2}$$