question_answer

In \[\Delta ABC,\] the slope of the median through A is \[-2,\,\,B=(-1,\,3)\] and \[C=(3,5)\]. If its area is 5, then the distance of the vertex A from the origin is

A)  2                                            

B)  4

C)  \[2\sqrt{2}\]                     

D)  \[3\sqrt{2}\]

Correct Answer: C

Solution :

 Let A = (p, q) and D = (1, 4) is the mid-point of BC. Slope of AD is - 2. \[\therefore \]  \[\frac{4-q}{1-p}=-2\] \[\Rightarrow \]               \[2p+q=6\]                         ?(i) Area of \[\Delta ABC\]\[=5\Rightarrow \,\left| \begin{matrix}    p & q & 1  \\    -1 & 3 & 1  \\    3 & 5 & 1  \\ \end{matrix} \right|=\pm \,10\] \[\Rightarrow \]               \[-2p+4q-14=\pm 10\] \[\Rightarrow \]               \[-p+2p-7=\pm \,5\] \[\Rightarrow \]               \[p-2q=-2,\,\,-12\]          ?(ii) \[\therefore \]  \[(p,\,q)=(0,\,6),\,(2,\,\,2)\] \[\therefore \]  \[OA=6,\,\,2\sqrt{2}\]