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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Q1: To find the equation of the tangent to the curve x^2 - xy - y^2 + 4 = 0 at point K(2,4), we first need to find the slope of the tangent using implicit differentiation. Step 1: Differentiate the equation with respect to x. (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: Isolate (dy)/(dx). 2x - y = (x + 2y) (dy)/(dx) (dy)/(dx) = (2x - y)/(x + 2y) Step 3: Substitute the coordinates of point K(2,4) into the derivative to find the slope m. m = (2(2) - 4)/(2 + 2(4)) = (4 - 4)/(2 + 8) = (0)/(10) = 0 Step 4: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), with m=0 and (x_1, y_1) = (2,4). y - 4 = 0(x - 2) y - 4 = 0 y = 4 The equation of the tangent is y=4. Q2: We are given the sum of the first n terms of an A.P. as S_n = n^2 - 8n. We need to find the least possible value of $