Question

Find the equation of the tangent line to the curve x³y³ = 6xy at the point (1, 6).

a) y = 2x - 4
b) y = 4x - 2
c) y = -2x + 4
d) y = -4x + 2

Answer 1

The equation of the tangent line to the curve x³y³ = 6xy at the point (1, 6) is y = 4x - 2 (Option b).

Step-by-step explanation:

To find the equation of the tangent line, we'll use the concept of implicit differentiation and the point-slope form of a line. First, implicitly differentiate the given curve equation with respect to x, yielding 3x²y³ + x³ ⋅ 3y²(dy/dx) = 6y + 6x(dy/dx). Then, solve for (dy/dx) and evaluate it at the given point (1, 6). Substituting this slope and the point into the point-slope form y - y₁ = m(x - x₁), where (x₁, y₁) is the given point, yields the equation y = 4x - 2, and the correct answer is (option b).

In summary, the process involves finding the derivative of the implicit function, determining the slope of the tangent line at the given point, and then using the point-slope form to derive the equation of the tangent line. The final equation y = 4x - 2 represents the line tangent to the curve x³y³ = 6xy at the point (1, 6). This method demonstrates the application of calculus concepts to analyze curves and determine tangent lines at specific points.