Final answer:
To find the points on the ellipse 4x² y² = 16 with a slope of 2, we can use derivative. The slope of the ellipse at any point can be found by differentiating y with respect to x. By equating the derivative to 2 and solving for x and y, we find that the two points are (-4, 1/4) and (4, -1/4).
Step-by-step explanation:
The equation of the given ellipse is 4x² y² = 16. To find the points on the ellipse with a slope of 2, we need to find the points where the derivative of y with respect to x is equal to 2.
We can start by rearranging the equation: y² = (4/16x²). Taking the derivative of both sides with respect to x, we get: 2y(dy/dx) = (4/16)(-8x⁻³). Simplifying, we have: 2y(dy/dx) = -1/x³. Dividing both sides by 2y, we find: dy/dx = -1/(2xy).
Now, equating dy/dx to 2, we have: 2 = -1/(2xy). Solving for y, we get: y = -1/(4x). Substituting this value of y into the original equation of the ellipse, we have: 4x²(-1/(4x))² = 16. Simplifying, we find: 1/x² = 16. Taking the square root of both sides and solving for x, we get two values: x = -4 and x = 4.
Substituting these values of x into the equation y = -1/(4x), we find the corresponding y-values: y = -1/(-4) = 1/4 and y = -1/(4) = -1/4.
Therefore, the two points on the ellipse with a slope of 2 are: (-4, 1/4) and (4, -1/4).