Question

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.

Answer 1

Let the point be P(x, y)

$$\therefore \left[\sqrt{{(x-4)}^2+{(y-0)}^2}\right]-\left[\sqrt{{(x+4)}^2+{(y-0)}^2}\right]=2$$

$$\Rightarrow {\left[\sqrt{{(x-4)}^2+{(y-0)}^2}\right]}^2={[2+\sqrt{{(x+4)}^2+{(y-0)}^2}]}^2$$

$$\Rightarrow {(x-4)}^2+y^2=4+{(x+4)}^2+y^2+4\sqrt{{(x+4)}^2+{(y-0)}^2}$$

$$\Rightarrow {(x-4)}^2-{(x+4)}^2=4+4\sqrt{{(x+4)}^2+{(y-0)}^2}$$

$$\Rightarrow -16x=4+4\sqrt{{(x+4)}^2+{(y-0)}^2}$$

$$\Rightarrow -16x-4=4\sqrt{{(x+4)}^2+{(y-0)}^2}$$

$$\Rightarrow -4(4x+1)=4\sqrt{{(x+4)}^2+{(y-0)}^2}$$

$$\Rightarrow -(4x+1)=\sqrt{{(x+4)}^2+{(y-0)}^2}$$

$$\Rightarrow 16x^2+8x+1=x^2+8x+16+y^2$$

$$\Rightarrow 15x^2-y^2=15$$

$$\Rightarrow \frac{x^2}{1}-\frac{y^2}{15}=1$$

Which is the equation of a hyperbola.