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Standard Equations of a Hyperbola

Let us consider that the transverse axis coincides with the x-axis.

If c is the distance from the centre to a foci, one focus will be at F1 = (- c, 0) and the other one at F2 = (c, 0). If suppose the constant difference of the distance from any point P(x, y) on the hyperbola to the foci F1 and F2 be 2a i.e., |PF1 - PF2| = 2a

PF1 - PF2 = 2a

Squaring both sides, we get
(x + c)2 + y2 = 4a2 ± 4a + (x - c)2 + y2
4xc = 4a2 4a

Squaring both the sides, we get
+ a2 - 2xc = x2 - 2xc + c2 + y2

x2 (c2 - a2) - a2y2 = a2 (c2 - a2)                                (1)
To get points on the hyperbola of the x-axis, we must have a<c. The reason for this is that if P is a point on the hyperbola (from the figure)

In triangle F1PF2, we must have
F1P < F2P + F1F2
F1P - F2P < F1F2

The difference of the distances from P, to the farther point minus the closer point be 2a. (i.e.,) F1P - F2P = 2a.

⇒ 2a < 2c or a < c.
Since 0 < a < c, then a2 < c2, so c2 - a2 > 0.
Let b2 = c2 - a2, b>0.
Dividing the equation (1) by a2 (c2 - a2), we get

This is an equation of the required hyperbola.

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