Question

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

Answer 1

`ax^2 + by^2 +cx + 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

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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