Final answer:
To find the equation of the tangent line for the given equation y = (x / (y + a)), we can differentiate the equation with respect to x using implicit differentiation and substitute the coordinates of the point of tangency. The equation of the tangent line is y = (1 / (1 + a)) * x + f.
Step-by-step explanation:
The equation of the tangent line can be found by taking the derivative of the given equation and substituting the coordinates of the point of tangency into the derivative equation. In this case, the given equation is y = (x / (y + a)). To find the derivative, we can use implicit differentiation.
First, rewrite the equation as xy + ay - x = 0. Now, differentiate both sides of the equation with respect to x. We will treat y as a function of x and apply the product rule: (x * dy/dx) + y + a * dy/dx -1 = 0. Simplifying this equation gives: (1 + a) * dy/dx = 1 - xy. Now, divide both sides by (1 + a) to solve for dy/dx:
dy/dx = (1 - xy) / (1 + a).
Now, substitute the given point of tangency (0, f) into the dy/dx equation. Since y = f at x = 0, we can substitute these values into the derivative equation:
dy/dx = (1 - (0)(f)) / (1 + a) = 1 / (1 + a).
Therefore, the equation of the tangent line in the form y = mx + b is y = (1 / (1 + a)) * x + f.