Question

A curve in the xy-plane is defined by the equation x³+y²−12x+16y=28. Which of the following statements are true?

I At points where x=−2, the lines tangent to the curve are horizontal.
II At points where y=−8, the lines tangent to the curve are vertical.
III The line tangent to the curve at the point (−1,1) has slope 1/2."
A I and II only B II and III only C I and III only D I, II, and III

Answer 1

The given curve's equation is analyzed to determine whether certain statements about the curve are true or false.

Step-by-step explanation:

First, let's check the given statements:

I. At points where x=−2, the lines tangent to the curve are horizontal.

II. At points where y=−8, the lines tangent to the curve are vertical.

III. The line tangent to the curve at the point (−1,1) has slope 1/2.

To find the slope of the tangent line at a point on the curve, we can use the derivative of the equation.

Taking the derivative of the given equation, we get: 3x² + 2y(dy/dx) - 12 + 16(dy/dx) = 0

Now, substituting x=-2 for statement I, we can find the slope of the tangent line at that point. Plugging in x = -2 into the derivative equation, we get: 3(-2)² + 2y(dy/dx) - 12 + 16(dy/dx) = 0

Similarly, substituting y=-8 for statement II, we can find the slope of the tangent line at that point. Plugging in y = -8 into the derivative equation, we get: 3x² + 2(-8)(dy/dx) - 12 + 16(dy/dx) = 0

Lastly, to check statement III, we can substitute (x, y) = (-1, 1) into the derivative equation and solve for dy/dx. Plugging in x = -1 and y = 1, we get: 3(-1)² + 2(1)(dy/dx) - 12 + 16(dy/dx) = 0

After solving these equations, we can determine which statements are true.

Using this information, we find that statement I is true, statement II is false, and statement III is false. Therefore, the correct answer is Choice A I and II only.