question_answer

Let \[A\equiv \left( -2,-2 \right)\] and \[B\equiv \left( 2,-2 \right)\] be two points and \[AB\] subtends an angle of \[45{}^\circ \] at any point \[P\] in the plane in such a way that area of \[\Delta APB\] is 8 square unit, then number of possible position (s) of P is

A) 1

B) 2

C) 4

D) infinity

Correct Answer: C

Solution :

\[\because AB=4\] and area of \[\Delta APB=8\]

\[\therefore \frac{1}{2}\times 4\times h=8\Rightarrow h=4,\]

Where, \[h\] is the height of \[\Delta APB\] from figure it is clear that \[P\] lies on circle of radius \[2\sqrt{2}\] unit with \[AB\] as its chord so there are four possible position of \[P.\]