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
Step 1: Differentiate the equation of the curve implicitly with respect to x. The given curve is x^2 + xy - y^2 + 4 = 0. Differentiate each term: (d)/(dx)(x^2) + (d)/(dx)(xy) - (d)/(dx)(y^2) + (d)/(dx)(4) = (d)/(dx)(0) Using the power rule, product rule for xy, and chain rule for y^2: 2x + (1 · y + x · (dy)/(dx)) - 2y (dy)/(dx) + 0 = 0 2x + y + x(dy)/(dx) - 2y(dy)/(dx) = 0 Step 2: Solve for (dy)/(dx). Group the terms containing (dy)/(dx): x(dy)/(dx) - 2y(dy)/(dx) = -2x - y Factor out (dy)/(dx): (dy)/(dx)(x - 2y) = -2x - y Isolate (dy)/(dx): (dy)/(dx) = (-2x - y)/(x - 2y) This can also be written as: (dy)/(dx) = (2x + y)/(2y - x) Step 3: Calculate the slope of the tangent at the point k(2, 4). Substitute x=2 and y=4 into the expression for (dy)/(dx): m = (2(2) + 4)/(2(4) - 2) m = (4 + 4)/(8 - 2) m = (8)/(6) m = (4)/(3) Step 4: Find the equation of the tangent line. Using the point-slope form of a linear equation, y - y_1 = m(x - x_1), with (x_1, y_1) = (2, 4) and m = (4)/(3): y - 4 = (4)/(3)(x - 2) Multiply both sides by 3 to eliminate the fraction: 3(y - 4) = 4(x - 2) 3y - 12 = 4x - 8 Rearrange the terms to the standard form Ax + By + C = 0: 0 = 4x - 3y - 8 + 12 4x - 3y + 4 = 0 The equation of the tangent to the curve at k(2, 4) is: 4x - 3y + 4 = 0 That's 2 down. 3 left today — send the next one.