Question

For waht value of $\lambda$ , the three numbers $2\lambda -1,\frac{1}{4},\lambda -\frac{1}{2}$ can be the direction cosines of a straight line?

• A. $\frac{1}{2}\pm \frac{\sqrt{3}}{4}$
• B. $\frac{3}{4}$
• C. $\pm \frac{3}{4}$
• D. $\frac{\sqrt{3}}{2}\pm \frac{1}{4}$

Answer 1

Answer: (A)

$\frac{1}{2}\pm \frac{\sqrt{3}}{4}$

for being direction cosines of straight line,
$(2\lambda -1{)}^2+{\left(\frac{1}{4}\right)}^2+{(\lambda -\frac{1}{2})}^2=1$
$4{\lambda }^2-4\lambda +y+\frac{1}{16}+{\lambda }^2-\lambda +\frac{1}{4}=\lambda$
$5{\lambda }^2-5\lambda +\frac{5}{16}=0$
$16{\lambda }^2-16\lambda +1=0$
$\lambda =\frac{16\pm \sqrt{256-64}}{32}$
$=\frac{16\pm \sqrt{13}}{32}=\frac{1}{2}\pm \frac{\sqrt{3}}{4}$