Question

If a line passing through $P(3,1)$ meets coordinate axes in $A$ and $B$ and distance from origin is maximum then area of $△OAB$ is

• A. $25$ sq. units
• B. $\frac{50}{3}$ sq. units
• C. $\frac{100}{3}$ sq. units
• D. $19$ sq. units

Answer 1

The correct option is B $\frac{50}{3}$ sq. units

Let the line be $y=mx+c⟹$ using P$(3,1)$ lies on the line $c=1-3m$ and $A(3-\frac{1}{m},0);B(0,1-3m)$ and let the perpendicular distance from origin to the line be p;

$p=\frac{|1-3m|}{\sqrt{1+m^2}}⟹m=-3⟹A(\frac{10}{3},0)B(0,10)$

Area of the $△OAB=\frac{1}{2}\frac{10}{\sqrt{10}}\frac{10\sqrt{10}}{3}=\frac{50}{3}$