This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to your questions. Q1) Find the equation of the tangent to the curve x^2 - xy - y^2 + 4 = 0 at K(2, 4). Step 1: Differentiate the equation implicitly with respect to x to find (dy)/(dx). (d)/(dx)(x^2) - (d)/(dx)(xy) - (d)/(dx)(y^2) + (d)/(dx)(4) = (d)/(dx)(0) 2x - (1 · y + x · (dy)/(dx)) - 2y (dy)/(dx) + 0 = 0 2x - y - x(dy)/(dx) - 2y(dy)/(dx) = 0 Step 2: Rearrange the equation to solve for (dy)/(dx). 2x - y = x(dy)/(dx) + 2y(dy)/(dx) 2x - y = (x + 2y)(dy)/(dx) (dy)/(dx) = (2x - y)/(x + 2y) Step 3: Evaluate the slope of the tangent at the point K(2, 4) by substituting x=2 and y=4. m = (2(2) - 4)/(2 + 2(4)) m = (4 - 4)/(2 + 8) m = (0)/(10) m = 0 Step 4: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), with (x_1, y_1) = (2, 4) and m=0. y - 4 = 0(x - 2) y - 4 = 0 y = 4 The equation of the tangent to the curve at K(2, 4) is y=4. That's 3 down. 2 left today — send the next one.