Final answer:
To find the value of x where the slope of the tangent is -1, we implicitly differentiate the equation x²y² = 36 and solve for dy/dx, which gives us x = ±6. Considering the negative slope, the correct answer is x = -6, not listed in the provided options.
Step-by-step explanation:
The student asks at what value of x does the tangent to the curve x²y² = 36 have a slope of -1. To find this, we first need to implicitly differentiate the given equation with respect to x, as follows:
2xy² + x²(2y)(dy/dx) = 0
Now, solving for dy/dx (the slope of the tangent), we get:
dy/dx = -y²/x²
We are looking for where the slope is -1. So we set dy/dx equal to -1 and solve for x:
-1 = -y²/x² => x² = y²
Since x²y² = 36, if x² = y², then x²x² = 36 => x² = √36 => x = ±6.
However, the slope is negative, meaning both x and y should have the same sign, hence we exclude x = 6 and choose x = -6 as the correct value where the slope of the tangent is -1.
The correct answer, considering possible options, is x = -6, which is not listed in the given options a, b, c, d.