Question

If the point $(\alpha ,\alpha )$ lies between the lines $|2x+y|=5$, then select one of the most appropriate option

• A. $\left|\alpha \right|<\frac{5}{3}$
• B. $\left|\alpha \right|<\frac{7}{2}$
• C. $\left|\alpha \right|<\frac{11}{3}$
• D. $\left|\alpha \right|<\frac{5}{2}$

Answer 1

The correct option is A $\left|\alpha \right|<\frac{5}{3}$

Given lines are
$|2x+y|=5$
i.e., $2x+y=\mathrm{5....}\left(i\right)$
or $2x+y=-\mathrm{5.....}\left(ii\right)$
Point $(\alpha ,\alpha )$ lie on the line
$y=x$
$\because$ Line $y=x$ intersect the line Eqs (i) and (ii) at points $(\frac{5}{3},\frac{5}{3})$ and $(\frac{-5}{3},\frac{-5}{3})$ respectively
(Here, we find point of intersects by solving line Eqs (i), (iii) and lines Eqs. (ii) and (iii))
$\therefore$ $\alpha$ lie in between $\frac{-5}{3}$ and $\frac{5}{3}$
i.e., $\frac{-5}{3}<\alpha <\frac{5}{3}$
$\therefore \left|\alpha \right|<\frac{5}{3}$