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

विकल्प

•  a hyperbola

•  a branch of the hyperbola

• a parabola

• an ellipse

Answer 1

 a hyperbola

\[\text { Let } P(x, y) \text { be any point on the hyperbola } \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 . \]

By definition, we have: 

\[PA = e\left( x - \frac{a}{e} \right) = ex - a\]

\[\text { and } PB = e\left( x + \frac{a}{e} \right) = ex + a\]

\[ \therefore PB - PA = \left( ex + a \right) - \left( ex - a \right) = 2a = k\]