1. Let y = f(x) be the continuous function that satisfies the equation x^4-5x^2y^2+4y^4=0 and whose graph contains the points (2, 1) and (-2, -2). Let l be the line tangent to the graph of f at x = 2.

Question

1.	Let y = f(x) be the continuous function that satisfies the equation x^4-5x^2y^2+4y^4=0 and whose graph contains the points (2, 1) and (-2, -2). Let l be the line tangent to the graph of f at x = 2. 
a.	Find an expression for y’ 
b.	Write an equation for line l

Steve · Answer

differentiate implicitly:

4x^3 - 10xy^2 - 10x^2yy' + 16y^3y' = 0
y'(16y^3 - 10x^2y) = 10xy^2 - 4x^3

y' = x(5y^2 - 2x^2) / y(8y^2 - 5x^2)
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y'(2) = 2(5-8) / (8-20) = -6/-12 = 1/2

Line l is (y-1)/(x-2) = 1/2