Q. 135.0( 2 Votes )

Show that the set

Answer :

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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