the temperature at each point x y of a metal

Question

The temperature at each point (x, y) of a metal plate is given by

T(x,y) = 1 + x² + 3/2y²

The path of a heat-seeking particle on the plate is a curve r(t) = (x(t), y(t)) such that for each t, the velocity e(t) of the particle at time t is in the direction of the fastest increase of the temperature Tat r(t). Suppose that particle starts at the point (1,1) with speed |r'(1)| = 8 units.

(a) Determine the equation for 9(t) thus obtaining differential equations for the functions x(t) and y(t). What initial conditions do these functions satisfy at t = 0?

(b) Use part (a) to find explicitly the functions x(t) and y(t) and thus the path r(t).