Question

A point moves in a plane so that its distances PA and PB from two fixed points A and B in the plane satisfy the relation PA − PB = k (k ≠ 0), then the locus of P is
(a) a hyperbola
(b) a branch of the hyperbola
(c) a parabola
(d) an ellipse

Answer 1

(a) a hyperbola
$$\begin{gathered} \mathrm{Let}P(x,y)\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{hyperbola}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1. \\ \mathrm{B}y\mathrm{definition},\mathrm{we}\mathrm{have}: \\ PA=e\left(x-\frac{a}{e}\right)=ex-a \\ \mathrm{and}PB=e\left(x+\frac{a}{e}\right)=ex+a \\ \therefore PB-PA=\left(ex+a\right)-\left(ex-a\right)=2a=k \end{gathered}$$