Question

If $l,m,n$ are the d.cs of a line and $l=\frac{1}{3}$, then the maximum value of $l\times m\times n$ is

• A. $4$
• B. $\frac{4}{9}$
• C. $\frac{27}{4}$
• D. $\frac{4}{27}$

Answer 1

The correct option is D $\frac{4}{27}$

Given that $l=\frac{1}{3}$
Now, $l^2+m^2+n^2=1$
$m^2+n^2=\frac{8}{9}$
Using A.M $\ge$ GM,
$\frac{m^2+n^2}{2}\ge \sqrt{m^2{n}^2}$
$\Rightarrow \frac{4}{9}\ge mn$
Therefore, $lmn\le \frac{4}{27}$