Let p and q be non-zero reals. The equation (p x² + q y² + r)(4x² + 4y² - 8x - 4) = 0 represents which of the following?

two straight lines and a circle, when r = 0 and p, q have opposite signs.
two circles, when p = q and r has sign opposite to p.
a hyperbola and a circle, when p and q have opposite signs and r is not 0.
a circle and an ellipse, when p and q are unequal but of the same sign and r has sign opposite to p.

Correct answer: a hyperbola and a circle, when p and q have opposite signs and r is not 0.

Solution

The factor 4(x² + y² - 2x - 1) = 0 is a circle. The factor p x² + q y² + r = 0 is a hyperbola when p and q have opposite signs and r != 0. So that pairing is correct.

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