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.
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You're on a roll — Step 1: Verify if the point K(2,4) lies on the curve. Substitute x=2 and y=4 into the equation of the curve x^2 - xy - y^2 + 4 = 0: (2)^2 - (2)(4) - (4)^2 + 4 4 - 8 - 16 + 4 -4 - 16 + 4 -20 + 4 = -16 Since -16 ≠ 0, the point K(2,4) does not lie on the curve. However, as the question asks for the tangent at this point, we will proceed by finding the slope of the curve at K(2,4) and using it to form the tangent line. This implies there might be a typo in the question, and we are treating K(2,4) as the point of tangency. Step 2: Differentiate the equation of the curve implicitly with respect to x. The equation 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) 2x - (1 · y + x · (dy)/(dx)) - 2y (dy)/(dx) + 0 = 0 2x - y - x(dy)/(dx) - 2y(dy)/(dx) = 0 Step 3: Solve for (dy)/(dx) and calculate the slope at K(2,4). Group terms containing (dy)/(dx): 2x - y = x(dy)/(dx) + 2y(dy)/(dx) 2x - y = (x + 2y)(dy)/(dx) (dy)/(dx) = (2x - y)/(x + 2y) Now, substitute the coordinates of point K(2,4) (x=2, y=4) into the derivative to find the slope m: m = (2(2) - 4)/(2 + 2(4)) m = (4 - 4)/(2 + 8) m = (0)/(10) m = 0 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=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. Send me the next one 📸