Question

A line passing through the point $(2,2)$ and the axes enclose an area $\lambda$.The intercepts on the axes made by the line are given by the two roots of

• A. $x^2-2\left|\lambda \right|x+\left|\lambda \right|=0$
• B. $x^2+\left|\lambda \right|x+3\left|\lambda \right|=0$
• C. $x^2-\left|\lambda \right|x+2\left|\lambda \right|=0$
• D. none of these.

Answer 1

Answer: (C)

$x^2-\left|\lambda \right|x+2\left|\lambda \right|=0$

Let line is $\frac{x}{a}+\frac{y}{b}=1$
which passes through $(2,2)$
Here, $a$ is the $x-$intercept and $b$ is the $y-$intercept.
So $\frac{2}{a}+\frac{2}{b}=1$ and $\frac{1}{2}ab=\lambda$
$\Rightarrow ab=2\lambda$
$\therefore \frac{2(a+b)}{ab}=1\Rightarrow a+b=\frac{ab}{2}\Rightarrow a+b=\lambda$
Product of roots is $2\lambda .$
and sum of roots is $\lambda .$