Question

The d.r.s of the line $AB$ are $6,-2,9$. lf the line $AB$ makes angles $\alpha ,\beta ,\gamma$ with $OX,OY,OZ$ respectively where $O=(0,0,0)$ then ${\sin }^2\alpha -{\sin }^2\beta =$

• A. $\frac{77}{121}$
• B. $\frac{-32}{121}$
• C. $77$
• D. $\frac{85}{121}$

Answer 1

Answer: (B)

$\frac{-32}{121}$

Given, DR's of $\overrightarrow{AB}=(6,-2,9)$

$\cos \alpha =\frac{\overrightarrow{AB}.\hat{i}}{\left|\overrightarrow{AB}\right|\left|\hat{i}\right|}=\frac{6}{\sqrt{6^2+2^2+9^2}}=\frac{6}{11}$

$\cos \beta =\frac{-2}{11}$

$\cos \gamma =\frac{9}{11}$

${\sin }^2\alpha -{\sin }^2\beta =(1-{\cos }^2\alpha )-(1-{\cos }^2\beta )$

$={\cos }^2\beta -{\cos }^2\alpha$

$=\frac{4}{121}-\frac{36}{121}=\frac{-32}{121}$