# Is the equation 4x^2-y^2-4x-3=0 a line, parabola, ellipse, hyperbola, or circle?

Nov 19, 2015

$a {x}^{2} + b {y}^{2} + c x + \mathrm{dy} + e = 0$

If both $a$ and $b$ are $0$, we are dealing with a line.

If either $a$ or $b$ (but not both) is $0$, then we are dealing with a parabola

If $a$ and $b$ have the same sign, then we are dealing with an ellipse

If $a$ and $b$ are equal (i.e. same sign and numerical value), then we are dealing with a circle

If $a$ and $b$ have different signs, then we are dealing with a hyperbola

In the given equation,

$a = 4$
$b = - 1$,

$a$ and $b$ are both non-zero but have different signs, we are dealing with a hyperbola