Final answer:
To find the area bounded by the given parametric curve and the fourth quadrant axes, identify the semi-major and semi-minor axis values from the equations, use the ellipse area formula, and divide by 4 for the fourth quadrant, which yields k = 1.
Step-by-step explanation:
The student is asking for the area of a plane figure bounded by a parametric curve and the coordinate axes in the fourth quadrant. The curve is given by the equations x = √2 × cos(t) and y = 4√2 × sin(t). To find this area, we would typically integrate the function over the interval corresponding to the fourth quadrant, which for the sine and cosine functions would mean an interval like [0, π/2] for t. However, without integral calculus, we can recognize that the described curve with these parametric equations represents an ellipse. For an ellipse with semi-major axis a and semi-minor axis b, the area is given by πab. The parametric equations gave us a = √2 and b = 4√2. The area in the fourth quadrant would be a quarter of the total ellipse, as each quadrant would contain an equal quarter section of the ellipse. Therefore, the area in the fourth quadrant is (π√2×4√2)/4 = π(√2)(√2)(4)/4 = π. The constant k that the student is looking for is thus 1.