Final answer:
To find the equations of the tangent lines to the curve y = x - 1/x that are parallel to the line x - 2y = 3, we need to find the derivative of the curve and set it equal to the slope of the given line. Solving for x and substituting back into the equation, we find two tangent lines: y = √2 - 1/√2 and y = -√2 + 1/√2.
Step-by-step explanation:
To find the equations of the tangent lines to the curve y = x - 1/x that are parallel to the line x - 2y = 3, we first need to find the derivative of the curve y = x - 1/x. The derivative is dy/dx = 1 + 1/x^2. Since the tangent lines we are looking for are parallel to the line x - 2y = 3, the slopes of the tangent lines must be equal to the slope of the given line. The slope of the given line is 1/2. Setting dy/dx equal to 1/2, we get 1 + 1/x^2 = 1/2. Solving for x, we find x = ±√2. Substituting these values of x back into the equation y = x - 1/x, we can find the corresponding y-values. Therefore, the equations of the tangent lines are y = √2 - 1/√2 and y = -√2 + 1/√2.