A curve C is given by the equation:

Question

A curve C is given by the equation:
x² + y² = (2x² + 2y² − x)²

(a) Find all points (0, y) on the curve C, where y≠0. 
(b) At each of the points found in the part (a) above, find an equation of the tangent line to C.

Answer 1

When x= 0, y = +/- 1/2. So find the equation of the tangent line at (0,1/2) and (0, -1/2).

 

Finding an explicit formula for y' using implicit differentiation is a formidable task. Instead just use implicit differentiation, leaving the result as you find it. Then substitute 0 for x, 1/2 for y, and solve for y'.

 

Something like this after implicit differentiation: 2x + 2yy' = 2(2x² + 2y² -x)(4x+4yy'-1).

 

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