Question

If the projection of the line segment $AB$ on the coordinate axes are $2,3,6$ then the square of the sines of the angles made by $AB$ with $OY$ axis is

• A. $\frac{3}{7}$
• B. $\frac{3}{49}$
• C. $\frac{4}{7}$
• D. $\frac{40}{49}$

Answer 1

Answer: (D)

$\frac{40}{49}$

Let length of $AB=k$ and angle made with $x,y$and $z$ axis be $\alpha ,\beta and\gamma$ respectively.

$\Rightarrow k\cos \alpha =2$

$k\cos \beta =3k\cos \gamma =6$

We know ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

$\Rightarrow \frac{4}{k^2}+\frac{9}{k^2}+\frac{36}{k^2}=1$

$\Rightarrow 49=k^2$

$\Rightarrow k=7$ units

$\therefore cos\beta =\frac{3}{7}$

${\sin }^2\beta =1-\frac{9}{49}=\frac{40}{49}$