Question

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

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

Answer 1

The correct option is B $|\alpha |<\frac{5}{3}$

$|2x+y|=5$
$\Rightarrow 2x+y=\pm 5$

Point $(\alpha ,\alpha )$ lies on the line $y=x$
$\therefore$ Intersection points of $y=x$ and $|2x+y|=4$ are
$(\frac{5}{3},\frac{5}{3})\text{and}(\frac{-5}{3},\frac{-5}{3})$
Point $(\alpha ,\alpha )$ lies in between the points
$(\frac{5}{3},\frac{5}{3})\text{and}(\frac{-5}{3},\frac{-5}{3})\therefore |\alpha |<\frac{5}{3}$


Alternate:
$(\alpha ,\alpha )$ lies inside the lines, $|2x+y|<5$
$\Rightarrow |2\alpha +\alpha |<5\Rightarrow |\alpha |<\frac{5}{3}$