Question

If $(2,-1,2)$ and $( K ,-3,-5)$ are the triads of direction ratios of two lines and the angle between the lines is $60^{\circ}$, then

• A. $K ^2-56 K -208=0$
• B. $5 K ^2-110 K +112=0$
• C. $7 K ^2-112 K -110=0$
• D. $7 K ^2-112 K +110=0$

Answer 1

Answer: (C)

$ a=(2 \hat{i}-\hat{j}+2 \hat{k}) $
$b=(k \hat{i}-3 \hat{j}-5 \hat{k}) $
$ a \cdot b=|a||b| \cos \theta $
$\because \cos 60^{\circ}=\frac{1}{2}$
$ (2 \hat{i}-\hat{j}+2 \hat{k})(k \hat{i}-3 \hat{j}-5 \hat{k})=3 \sqrt{k^2+34}\left(\frac{1}{2}\right)$
$ 2(2 k-7)=3 \sqrt{k^2+34}$
$ \Rightarrow 4\left(4 k^2+49-28 k\right)=9\left(k^2+34\right) $
$ 7 k^2-112 k-110=0$