Show that the set of all points such that the difference of their distances from (4, 0) and (-4, 0) is always equal to 2 represents a hyperbola.

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