question_answer

The coordinates of the points O, A and B are (0,0), (0,4) and (6,0) respectively. If a points P moves such that the area of \[\Delta POA\]is always twice the area of \[\Delta POB\], then the equation to both parts of the locus of P is [IIT 1964]

A) \[(x-3y)(x+3y)=0\]

B) \[(x-3y)(x+y)=0\]

C) \[(3x-y)(3x+y)=0\]

D) None of these

Correct Answer: A

Solution :

The three given points are \[O\,\,(0,\,\,0),\,\,A(0,\,\,4)\] and \[B\,(6,\,\,0)\] and let \[P(x,\,\,y)\] be the moving point. Area of \[\Delta POA=2\,.\,\]Area of \[\Delta POB\] \[\Rightarrow \,\,\frac{1}{2}\times 4\times x=\pm \,2\times \frac{1}{2}\times 6\times y\] or \[x=\pm \,3y\] Hence the equation to both parts of the locus of P is \[(x-3y)\,(x+3y)=0\].