Find the equation of the locus of all points such that the difference of their distances from (4,0) and (-4,0) is always equal to 2. Find out

Dear Student,

We need to find the equation of the locus of all points such that the difference of their distances from (4,0) and (-4,0) is always equal to 2.
$$\begin{gathered} \Rightarrow \sqrt{{(x-4)}^2+{(y-0)}^2} - \sqrt{{(x+4)}^2+{(y-0)}^2} = 2 \\ \Rightarrow \sqrt{{(x-4)}^2+y^2} = 2+\sqrt{{(x+4)}^2+y^2} \\ Squaring both sides \\ \Rightarrow {(x-4)}^2+y^2=4+{(x+4)}^2+y^2+4\sqrt{{(x+4)}^2+y^2} \\ \Rightarrow x^2-8x+16+y^2=4+x^2+8x+16+y^2+4\sqrt{{(x+4)}^2+y^2} \\ \Rightarrow -16x-4=4\sqrt{{(x+4)}^2+y^2} \\ \Rightarrow -4(4x+1)=4\sqrt{{(x+4)}^2+y^2} \\ \Rightarrow -(4x+1)=\sqrt{{(x+4)}^2+y^2} \\ Squaring both sides \\ \Rightarrow 16x^2+8x+1=x^2+8x+16+y^2 \\ \Rightarrow 15x^2-y^2=15 \\ Dividing both sides by 15 \\ \Rightarrow \frac{x^2}{1}-\frac{y^2}{15}=1 \end{gathered}$$
Which is clearly the equation of parabola.

Regards