Question

Through the point $(1,1)$, a straight line is drawn so as to form with the coordinate axes a triangle of area $S$. The intercepts made by the line on the axes are the roots of the equation

• A. $x^2-\left|S\right|x+2\left|S\right|=0$
• B. $x^2+\left|S\right|x+2\left|S\right|=0$
• C. $x^2-2\left|S\right|x+2\left|S\right|=0$
• D. none of these

Answer 1

The correct option is C $x^2-2\left|S\right|x+2\left|S\right|=0$

If $a,b$ are the intercept made by the line, then the equation of the line is $\frac{x}{a}+\frac{y}{b}=1$

Since it passes through $(1,1)$

$\therefore \frac{1}{a}+\frac{1}{b}=1\Rightarrow \frac{a+b}{ab}=1$ ....(1)

Also, area of the triangle made by the straight line on the coordinate axes is $S$

$\therefore \frac{1}{2}ab=\left|S\right|\Rightarrow ab=2\left|S\right|$ ...(2)

So, by (1), $a+b=2|S|$ ...(3)

FRom (2) and (3) , the intercept $a$ and $b$ are the roots of the equation $x^2-2\left|S\right|x+2\left|S\right|=0$.