A straight line passing through `P(3,1)` meets the coordinate axes at `A` and `B`. It is given that the distance of this straight line from the origin `O` is maximum. The area of triangle `O A B` is equal to

Options

A. `50//3` sq.units
B. `25//3` sq.units
C. `20//3` sq.units
D. `100//3` sq.units

Answer

A

Explanation

Line AB will be the farthest from the origin if OP is right angled to the line drawn.
`OP=sqrt10`
Also, ` tan theta=(1)/(3)`
`OA =OPsectheta`
`=sqrt(10)xx(sqrt(10))/(3)=10`
`OB=OP cosectheta=sqrt(10)xx(sqrt(10))/(1)=10`
`therefore` Area of `DeltaOAB=(1)/(2)=(1)/(2)(10/(3)(10)=(50)/(3)`