Determine the points

Question

Determine the points where the instantaneous rate of change dy/dx of the equation y² - 2x² = 32 is equal to 1.

Answer 1

Take the derivative explicitly

So solving for dy/dx we get dy/dx = 2x/y but dy/dx = 1 so 1 = 2x/y or y = 2x

Then, to find points on the curve that comply, we can plug in 2x in place of y so:

(2x)² - 2x² = 32

solving for x, we get x = ±4

Plugging the x back into the equation to find the y, for x = 4:

y² - 2(4)² = 32 or y = ±8 so the two points associated with that are (4, 8) and (4, -8)

For x = -4:

y² - 2(-4)² = 32 or y = ±8 so the two points associated with that are (-4, 8) and (-4, -8)

So now we have 4 points: (4, 8) (4, -8) (-4, 8) and (-4, -8)

Double checking on the original derivative dy/dx = 2x/y we see that only (4, 8) and (-4, -8) will give dy/dx = 1 (the others give dy/dx = -1) so the two points are (4, 8) and (-4, -8)

Answer 2

y² = 2x²+ 32

2y (dy/dx) = 4x

dy/dx = 2x/y = 2x/sqrt(2x² + 32) = 1

4x² = 2x² + 32

2x² = 32

x=±4 and I will leave determination of the ordinates to you!