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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

Asked by maria v 24th February 2015, 4:09 PM
Answered by Expert
Answer:

L e t space P left parenthesis x comma y right parenthesis space b e space a n y space p o i n t. G i v e n space t h a t space t h e space d i f f e r e n c e space o f space t h e i r space d i s tan c e s space f r o m space left parenthesis 4 comma 0 right parenthesis space a n d space left parenthesis negative 4 comma 0 right parenthesis space i s space 2. T h a t space i s comma square root of open parentheses x minus 4 close parentheses squared plus y squared end root minus square root of open parentheses x plus 4 close parentheses squared plus y squared end root equals 2 rightwards double arrow square root of open parentheses x minus 4 close parentheses squared plus y squared end root equals 2 plus square root of open parentheses x plus 4 close parentheses squared plus y squared end root rightwards double arrow open parentheses x minus 4 close parentheses squared plus y squared equals open square brackets 2 plus square root of open parentheses x plus 4 close parentheses squared plus y squared end root close square brackets squared S i m p l i f y i n g comma space w e space g e t comma minus 8 x equals 4 plus 8 x plus 4 square root of open parentheses x plus 4 close parentheses squared plus y squared end root rightwards double arrow negative 16 x minus 4 equals 4 square root of open parentheses x plus 4 close parentheses squared plus y squared end root rightwards double arrow open parentheses negative 16 x minus 4 close parentheses squared equals open square brackets 4 square root of open parentheses x plus 4 close parentheses squared plus y squared end root close square brackets squared A f t e r space s i m p l i f i c a t i o n comma space w e space h a v e comma 256 x squared plus 16 plus 128 x equals 16 x squared plus 256 plus 128 x plus 16 y squared rightwards double arrow 256 x squared plus 16 minus 16 x squared minus 256 minus 16 y squared equals 0 rightwards double arrow 240 x squared minus 240 minus 16 y squared equals 0 T h u s space t h e space l o c u s space o f space t h e space p o i n t s space i s space a space h y p e r b o l a.

Answered by Expert 24th February 2015, 6:22 PM
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