Final answer:
The correct statement about curve C defined by x²y=4 at the point (2,1) is that it is symmetric with respect to the y-axis. The slope of the tangent line at this point is -1, not 1, and the curve does not pass through the point (1, 2), nor is it symmetric with respect to the x-axis.
Step-by-step explanation:
Let C be the curve defined by x²y=4. We are evaluating the curve C at the point (2,1). To find the slope of the tangent at the point (2,1), we need to take the derivative of the given function with respect to x.
The differentiation of x²y with respect to x using implicit differentiation gives us 2xy + x²(dy/dx) = 0. Solving for dy/dx (the slope of the tangent), we find dy/dx = -(2xy) / x².
Plugging in the point (2,1) into this derivative gives us dy/dx = -(2*2*1) / (2²), which simplifies to -1. So, the slope of the tangent line at the point (2,1) is -1, not 1, making option A incorrect.
If we evaluate the given equation at the point (1, 2), we get (1)² * 2 = 2, which does not equal 4. Therefore, the curve does not pass through the point (1, 2), making option B incorrect.
To determine symmetry, we replace x with -x and y with -y to see if the equation remains unchanged. Replacing x with -x in x²y=4, we get (-x)²y = 4, which simplifies to x²y=4, the same as our original equation. Hence the curve is symmetric with respect to the y-axis, making option C correct.
Replacing y with -y, we get x²*(-y) != 4, so the curve is not symmetric with respect to the x-axis, making option D incorrect.