# Show that the set

To prove: the set of all points under given conditions represents a hyperbola

Let a point P be (x, y) such that the difference of their distances from (4, 0) and (-4, 0) is always equal to 2.

Formula used:

The distance between two points (m, n) and (a, b) is given by

The distance of P(x, y) from (4, 0) is

The distance of P(x, y) from (-4, 0) is

Since, the difference of their distances from (4, 0) and (-4, 0) is always equal to 2

Therefore,

Squaring both sides:

Squaring both sides:

16x2 + 1 – 8x = (x – 4)2 + y2

16x2 + 1 – 8x = x2 + 16 – 8x + y2

16x2 + 1 – 8x – x2 – 16 + 8x – y2 = 0

15x2 – y2 – 15 = 0

Hence, required equation of hyperbola is 15x2 – y2 – 15 = 0

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