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) =√(x−4)2+y2 Distance P(x,y) from (−4,0) =√(x+4)2+y2 Difference between distance=2 =√(x−4)2+y2−=√(x+4)2+y2 =√(x−4)2+y2−=2+√(x+4)2+y2 Squaring both sides,we get (x−4)2+y2=4+4=√(x+4)2+y2+(x+4)2+y2 (x−4)2+y2−(x+4)2−y2=4+4√(x+4)2+y2 (x−4−x−4)(x−4+x+4) =4+4√(x+4)2+y2 \ (-16x-4=4\sqrt{(x+4)^2+y^2}\) −4x−1=√(x+4)2+y2 Squaring both sides,we get 16x2+8x+1=x2+8x+16+y2 15x2−y2=15 This is hyperbola.

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