Given: a particle moves along the curve y = x² + 2x.

To find the points at which the curve are the x and y coordinates of the particle changing at the same rate

Equation of curve is y = x² + 2x

Differentiating the above equation with respect to x, we get

When x and y coordinates of the particle are changing at the same rate, we get

Now substitute the value from eqn(i), we get

2x + 2 = 1 ⇒ 2x = - 1

Substitute this value of x in the given equation of curve, we get

y = x² + 2x

Hence the points at which the curve are the x and y coordinates of the particle changing at the same rate is