If x and y are functions of time (t), the derivative with respect to t is

2*16*x*x' + 2*9*y*y' = 0

If x' = ±y', then

±16x + 9y = 0

x = ±9y/16

Plugging this into the equation for the ellipse, we get

16*(9/16y)^2 + 9*y^2 = 144

y^2 = 144/(9^2/16+9) = 10.24 = 3.2^2

y = ±3.2

x = ±9/16*y = ±1.8

The four points where the slope is ±1 are

If you insist that the rates of change actually be equal, rather than just of equal magnitude, then this will only occur at the two points

(x, y) = (-1.8, 3.2) or (1.8, -3.2),

that is, those points where the slope is +1.

2*16*x*x' + 2*9*y*y' = 0

If x' = ±y', then

±16x + 9y = 0

x = ±9y/16

Plugging this into the equation for the ellipse, we get

16*(9/16y)^2 + 9*y^2 = 144

y^2 = 144/(9^2/16+9) = 10.24 = 3.2^2

y = ±3.2

x = ±9/16*y = ±1.8

The four points where the slope is ±1 are

**(±1.8, ±3.2)**.If you insist that the rates of change actually be equal, rather than just of equal magnitude, then this will only occur at the two points

(x, y) = (-1.8, 3.2) or (1.8, -3.2),

that is, those points where the slope is +1.