Question

The point P(x,y) is moving along the curve  in such a way that the rate of change  of y is constant. Find the values of x at the points at which the rate of change of x is equal to half the rate of change of y.

Solution

We are required to find values of x at the points for which;

Let , then;

Rate of change of  with respect to  is;

Therefore;

We are given that;

Therefore;

Rule for differentiation of  is:

Rule for differentiation of  is:

Hence;

We recognize that it is a quadratic equation.

Standard form of quadratic equation is;

The solution of this quadratic equation is;

Therefore, for the given equation;

Now we have two options.