Final answer:
To determine the equation of the tangent line to the curve at (1,1), one must first differentiate the curve equation with respect to x, calculate the slope of the tangent at (1,1), and then apply the point-slope formula using this value.
Step-by-step explanation:
To find the equation of the tangent line to the curve at the given point (1,1), we'll need to calculate the derivative of the curve, identify the slope at the point, and then use the point-slope form to write the equation of the tangent. Starting with the given equation 13x - xy + y^2 = 13, first, we differentiate both sides with respect to x, applying the product rule where needed.
After differentiating, we plug in the coordinates of the given point (1,1) to find the slope of the tangent line. The slope (denoted as m) at the point (1,1) will be the value of the derivative at that point. Lastly, we use the point-slope equation y - y1 = m(x - x1) to write the equation of the tangent line.