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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 $2$ represent a hyperbola.

1 Answer

  • Distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
It is given that the difference of the distance between the points $(4,0)$ and $(-4,0)$ = 2
Hence $\sqrt{(x-4)^2+(y-0)^2}-\sqrt{(x+4)^2+(y-0)^2}=2$
$\Rightarrow \sqrt{(x-4)^2+y^2}=2+\sqrt{(x+4)^2+y^2}$
Squaring on both sides we get,
On expanding we get,
$\Rightarrow -(4x+1)=\sqrt{(x+4)^2+y^2}$
Squaring on both sides we get,
Dividing throughout by 15 we get,
This is clearly the equation of a hyperbola.
Hence proved.
answered Oct 14, 2014 by sreemathi.v

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