Parabola

To solve the problem of finding the point on the curve y² = 16x where the y-coordinate changes twice as fast as the x-coordinate, we follow the derivative approach:

1. Given the equation of the curve: y² = 16x, differentiate both sides with respect to t (time) to understand how y and x change.

2. Applying implicit differentiation: 2y(dy/dt) = 16(dx/dt).

3. Given that the y-coordinate changes twice as fast as the x-coordinate, we have: dy/dt = 2(dx/dt).

4. Substitute dy/dt = 2(dx/dt) into the differentiated equation:
2y(2(dx/dt)) = 16(dx/dt).

5. Simplify and cancel out dx/dt (assuming it is not zero): 4y = 16.

6. Solve for y: y = 4.

7. Substitute y = 4 back into the original curve equation y² = 16x: 16 = 16x.

8. Solve for x: x = 1.

Therefore, the point on the curve is (1, 4).