The curve 2(x^2 + y^2)^2 = 25(x^2 - y^2) is called a Lemniscates.
Find the tangent line at (3,1).
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Since y is not given as an explicit function of x here, it would be best not to try and find y' as an explicit function of x. Taking the derivative ...
This shows how to find the tangent line to a Lemniscates curve.