Question

Let L be the tangent line to the curve x^(2/3) + y^(2/3) = 4 at the point (−3√3, 1). Find the area of the triangle bounded by L and the axes.

A) 9 square units
B) 6 square units
C) 12 square units
D) 15 square units

Answer 1

To calculate the area of the triangle bounded by the tangent line L and the axes, we first find the slope of the tangent to the curve at the given point, then write the equation for L, and use the x-intercept and y-intercept to determine the area using the formula for the area of a right triangle.

Step-by-step explanation:

We need to find the area of the triangle bounded by the tangent line L and the axes. First, we determine the slope of the line L by differentiating the given curve x(2/3) + y(2/3) = 4. After finding the slope at the point (-3√3, 1), we use the point-slope form to write the equation of the tangent line L.

Next, we find the x-intercept and y-intercept of the line L to define the right triangle. The area of a right triangle is equal to 1/2 × base × height. Here, the base and height will be the absolute values of the x-intercept and y-intercept, respectively.

After calculating the intercepts, the formula for the area of a triangle gives us the solution to the problem. Finally, we compare the calculated area with the options provided to find the correct answer.