Discussion
The characteristic curve and the envelope of the one parameter family of surfaces
\[ (x - a)^{2} + (y - 2a)^{2} + z^{2} = 1 \]
are :
Answer: A
The characteristics are great circles of the spheres and the envelope is (y − 2x)2 + 5z2 = 5.
The family is spheres of radius 1 centred at \((a, 2a, 0)\). Differentiate \(F = (x - a)^{2} + (y - 2a)^{2} + z^{2} - 1\) with respect to a:
\[ \begin{aligned} -2(x - a) - 4(y - 2a) &= 0 \\ a &= \frac{x + 2y}{5} \end{aligned} \]
This plane \(x + 2y = 5a\) passes through the centre \((a, 2a, 0)\), so it cuts each sphere in a great circle: that is the characteristic curve. Substituting a back, \(x - a = \frac{2(2x - y)}{5}\) and \(y - 2a = \frac{y - 2x}{5}\), so
\[ \begin{aligned} \frac{4(y - 2x)^{2} + (y - 2x)^{2}}{25} + z^{2} &= 1 \\ (y - 2x)^{2} + 5z^{2} &= 5 \end{aligned} \]
The envelope is this cylinder, not \((y - 2x)^{2} + 2z^{2} = 2\) and not a pair of planes.
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