Final Answer:
The equation of the tangent line to the curve x² + 2xy - y²x = 17 at the point (3, 5) is b. y = (4/3)x + 23/3.
Step-by-step explanation:
To find the equation of the tangent line, we use implicit differentiation on the given curve x² + 2xy - y²x = 17. Differentiating each term with respect to x and substituting the given point (3, 5) yields the slope of the tangent line. The result is y' = dy/dx = 4/3. Using the point-slope form y - y₁ = m(x - x₁) with m = 4/3, x₁ = 3, and y₁ = 5, we obtain the equation of the tangent line as y = (4/3)x + 23/3.
Implicit differentiation involves differentiating both sides of an equation with respect to x. In this case, it's applied to the equation x² + 2xy - y²x = 17. By finding the first derivative, we get 2x + 2y + 2xy' - y² - y = 0. Solving for y' and substituting the point (3, 5) gives y' = 4/3. This slope is used in the point-slope form of the equation of the tangent line.
In conclusion, the application of implicit differentiation allows us to find the slope of the tangent line to the curve at the given point. Using this slope and the given point, we can derive the equation of the tangent line, resulting in the correct answer y = (4/3)x + 23/3.