given the curve x+xy+2y^2=6 a. find an expression for the slope at any point (x,

Question

given the curve x+xy+2y^2=6 a. find an expression for the slope at any point (x,y) on the curve b. write an equation for the line tangent to the curve at the point (2,1) c. find the coordinates of all other points on this curve with slope equal to the slope at (2,1)

Explanation / Answer

x + xy + 2y^2 = 6 a. Use implicit differentiation: 1 + y + x dy/dx + 4y dy/dx = 0 (x + 4y) dy/dx = -y - 1 dy/dx = (-y - 1) / (x + 4y) _________________________ b. At point (2,1) ---> dy/dx = (-1-1)/(2+4) = -2/6 = -1/3 Tangent line: y - 1 = -1/3 (x - 2) y = -1/3 x + 2/3 + 1 y = -1/3 x + 5/3 c. When x = 3: x + xy + 2y^2 = 6 3 + 3y + 2y^2 = 6 2y^2 + 3y - 3 = 0 y = (-3 ± v33)/4 y > 0 -----> y = (v33-3)/4 At point (3, (v33-3)/4) dy/dx = (-(v33-3)/4 - 1) / (3 + (v33-3)) dy/dx = ((-v33+3-4)/4) / v33 dy/dx = (-v33 - 1) / (4v33) Slope of normal = (4v33)/(v33+1) = (4v33)(v33-1)/((v33+1)(v33-1)) = (132-4v33)/(33-1) = (132-4v33)/32 = (33-v33)/8 Equation of normal line with slope (33-v33)/8 passing through point (3, (v33-3)/4) y - (v33-3)/4 = (33-v33)/8 (x - 3) y = (33-v33)/8 x + (-99+3v33)/8 + (v33-3)/4 y = (33-v33)/8 x + (5v33 -105)/8 _________________________ d. Slope at (2,1) = -1/3 dy/dx = -1/3 (-y - 1) / (x + 4y) = -1/3 -3(-y - 1) / (x + 4y) = 1 3y + 3 = x + 4y 3 - y = x Equation of curve: x + xy + 2y^2 = 6 (3-y) + (3-y)y + 2y^2 = 6 3 - y + 3y - y^2 + 2y^2 - 6 = 0 y^2 + 2y - 3 = 0 (y - 1) (y + 3) = 0 y = 1, y = -3 y = 1 -----> x = 3 - y = 2 y = -3 ----> x = 3 - y = 6 The other point with slope equal to slope at (2,1) is (6, -3) _________________________ _________________________ Note that solution for (c) involves some not-so-simple calculations. Maybe problem asks for normal to curve at point when x = -3, y > 0? When x = -3: x + xy + 2y^2 = 6 -3 - 3y + 2y^2 = 6 2y^2 - 3y - 9 = 0 (2y + 3) (y - 3) = 0 y > 0 -----> y = 3 At point (-3, 3) dy/dx = (-y - 1) / (x + 4y) = (-3 - 1) / (-3 + 12) = -4/9 Slope of normal = 9/4 Equation of normal line with slope 9/4 passing through point (-3,3) y - 3 = 9/4 (x + 3) y = 9/4 x + 27/4 + 3 y = 9/4 x + 39/4

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