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

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Answer 1 · 2015-11-19T05:37:40

$a x^{2} + b y^{2} + c x + d y + e = 0$ $ax^2 + by^2 +cx + dy + e = 0$

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

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

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

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

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

In the given equation,

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

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

Is the equation 4x2−y2−4x−3=04x^2-y^2-4x-3=0 a line, parabola, ellipse, hyperbola, or circle?