The line segment connecting $(-2,7)$ and $(3,11)$ can be parameterized by the equations
\begin{align*} x &= at + b, \\ y &= ct + d, \end{align*}
where $0 \le t \le 1,$ and $t = 0$ corresponds to the point $(-2,7).$ Find $a^2 + b^2 + c^2 + d^2.$
The parametric equations are x = 3t - 2 and y = 5t + 7, so a^2 + b^2 + c^2 + d^2 = 3^2 +(-2)^2 + 5^2 + 7^2 = 87.
