If the angle between the lines whose direction ratios are 2,-1 , … - Vidyadip

MCQ

If the angle between the lines whose direction ratios are $2,-1 , 2$ and $a, 3, 5$ be $45^\circ $, then $a =$

A
$1$
B
$2$
C
$3$
✓
$4$

✓

Answer

Correct option: D.

$4$

d
(d) $\cos \theta = \frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}$

$\frac{1}{{\sqrt 2 }} = \frac{{2a - 3 + 10}}{{\sqrt {{1^2} + {2^2} + {1^2}} \,\sqrt {{a^2} + {3^2} + {5^2}} }}$

$ \Rightarrow \,\,9\,({a^2} + 34) = 2\,{[2a + 7]^2} = \,2\,[4{a^2} + 28a + 49]$

$ \Rightarrow \,\,{a^2} - 56a + 208 = 0\,\,$

$\Rightarrow \,\,a = 4$.

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