Nonlinear Equations and Functions Question

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For the hyperbola centered at the origin with foci

$F_1 = (-c,0)$ and

$F_2=(c,0)$ , derive the equation of the hyperbola.

Grade 11 Nonlinear Equations and Functions

Squaring both sides, expanding, and combing like terms, dividing by any common constants, and putting all terms without a square root on the left side of the equation results in which of the following equations?

$2cx - a = 0$
$cx pm a^2 = -2 sqrt((x-c)^2 + y^2)$
$cx - a^2 = 0$
$cx - a^2 = pm a sqrt( (x-c)^2 + y^2)$