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.

Answer 1

$\text{Let P(x,y)be a point of the set.}$

$\text{Distance of P(x,y)from (4,0)}$

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

$DistanceP(x,y)from(-4,0)$

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

$\text{Difference between distance=2}$

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

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

$\text{Squaring both sides,we get}$

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

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

$(x-4-x-4)(x-4+x+4)$

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

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

$-4x-1=\sqrt{(x+4{)}^2+y^2}$

$\text{Squaring both sides,we get}$

$16x^2+8x+1=x^2+8x+16+y^2$ $15x^2-y^2=15$ This is hyperbola.