Question

Among the lines passing through $C(3,1)$, BA is farthest from the origin and cuts x-axis and y-axis at A and B respectively (where O is origin), then which of he following is/are incorrect.

• A. Equation of AB is $3x-y=10$
• B. $|AB|=\frac{10\sqrt{10}}{3}$
• C. Equation of AB is $3x+y=10$
• D. Area of $\Delta AOB=\frac{50}{3}$ sq.units

Answer 1

The correct option is A Equation of AB is $3x-y=10$

If BA is the farthest line from the origin then it must be perpendicular to the line joining origin and the point $(3,1)$

The line joining origin and the point $(3,1)$ is $\frac{y}{x}=\frac{1}{3}⟹3y=x$

The slope of the line BA is $-3$

Therefore the line BA is given by $\frac{y-1}{x-3}=-3⟹y=-3x+10$

Line BA is $3x+y=10$

and points A and B are $(10/3,0)$ and $(0,10)$

Therefore $|AB|=\sqrt{{10}^2+(10/3{)}^2}=\frac{10\sqrt{10}}{3}$ units

Area of the $△AOB=\frac{1}{2}\times 10\times \frac{10}{3}=\frac{50}{3}$ sq.units