Answer
$1/4$
Work Step by Step
Step 1. Given $y=x^2+C$, we have $y'=2x+C'$ which gives the slope of tangent lines to the curve.
Step 2. As the line $y=x$ is a tangent to the curve, we have $2x+C'=1$ or $x=(1-C')/2$
Step 3. The intersect point between the line and the parabola is given by $x^2+C=x$ or $C=x-x^2$
Step 4. We assume $C$ is a constant; then $C'=0$ and $x=1/2$ from step 2.
Step 5. The results from step 3 gives $C=1/2-(1/2)^2=1/4$
Note: if $C$ is not a constant, we will need to solve a differential equation.
