31.If the angle between the lines whose direction ratios are 2,-1,2 and a,3,5 be `45^@`, then a =(A) 1(B) 2(C) 3(D) 4

Options

A. 52
B. 4
C. 2
D. 1

Answer

B

Explanation

Given direction ratios are (2,-1,2) and (x,3,5) We know that the angle between the lines whose direction ratios are `(a_1,b_1,c_1) and (a_2,b_2,c_2)` is
`cos theta =(a_1a_2+b_1b_2+c_1c_2)/(sqrt(a_1^2+b_1^2+c_1^2)sqrt(a_2^2+b_2^2+c_2^2))`
`rArr cos .pi/4=(2x-3+10)/(sqrt(4+1+4)sqrt(x^2+9+25))=(2x+7)/(sqrt9.sqrt(x^2+34))`
`rArr 1/sqrt2=(2x+7)/(3sqrt(x^2+34))rArr 2x+7=3sqrt((x^2+34)/2)`
`rArr4x^2=49+28x=(9(x^2+34))/2` (Squaring on both sides)
`rArr 2(4x^2+49+28x)=9x^2+306`
`rArr 8x^2+98+56x=9x^2+306`
`8x^2+98+56x=9x^2+306`
`x^2-56x+208=0`
`:. x=(56pmsqrt(3136-812))/2=(56pm48)/2=28pm24`
= 4,52.
Smaller value = 4.