Please help me by verifying my answer. I'm trying to teach myself here and I don't know if I got this whole envelope to family of curves thing right.
Determine the envelope to the family of curves: $ (1-C^2)x + 2Cy -a =0 $ ,where C is a real parameter and a is a real constant.
So the first thing I've done is find the partial derivative in respect to C of this equation: $ -2Cx + 2y $
Next I've formed a system with the original equation and the new derivative. From that system I got that $ C = \sqrt{\frac{a-x}{x}} $
To find the final implicit equation I've replaced C in the first original equation:
$2x-2a+2y\sqrt{\frac{a-x}{x}} = 0$
Is this how it is supposed to work out? If not, please explain to me how to get to the desired result both by an example and theoretically. Thank you in advance!