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Question

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


Solution

Let P(x,y)be a point of the set.

Distance of P(x,y)from (4,0)

=(x4)2+y2

Distance P(x,y) from (4,0)

=(x+4)2+y2

Difference between distance=2

=(x4)2+y2=(x+4)2+y2

=(x4)2+y2=2+(x+4)2+y2

Squaring both sides,we get

(x4)2+y2=4+4=(x+4)2+y2+(x+4)2+y2

(x4)2+y2(x+4)2y2=4+4(x+4)2+y2

(x4x4)(x4+x+4)

=4+4(x+4)2+y2 \

(-16x-4=4\sqrt{(x+4)^2+y^2}\)

4x1=(x+4)2+y2

Squaring both sides,we get

16x2+8x+1=x2+8x+16+y2 15x2y2=15 This is hyperbola.

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