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Question: The direction ratios of the line \(x - y + z - 5 = 0 = x - 3y - 6\)are proportional to \( {\te...

The direction ratios of the line x−y+z−5=0=x−3y−6x - y + z - 5 = 0 = x - 3y - 6are proportional to
a. 3,1,−2 b. 2,−4,1 c. 314,114,−214 d. 241,−441,141  {\text{a}}{\text{. }}3,1, - 2 \\\ {\text{b}}{\text{. }}2, - 4,1 \\\ {\text{c}}{\text{. }}\dfrac{3}{{\sqrt {14} }},\dfrac{1}{{\sqrt {14} }},\dfrac{{ - 2}}{{\sqrt {14} }} \\\ {\text{d}}{\text{. }}\dfrac{2}{{\sqrt {41} }},\dfrac{{ - 4}}{{\sqrt {41} }},\dfrac{1}{{\sqrt {41} }} \\\

Explanation

Solution

Hint: - Individual multiplication of direction ratios are zero so let’s assume them and write their multiplication with direction vectors of other equation equal to zero.

Let the direction ratios be l,m,nl,m,n
So, in terms of vector it is written as
a⃗=li^+mj^+nk^\vec a = l\hat i + m\hat j + n\hat k
Given equations are
x−y+z−5=0 x−3y−6=0  x - y + z - 5 = 0 \\\ x - 3y - 6 = 0 \\\
So, the direction ratios of first equation is (1,−1,1)\left( {1, - 1,1} \right)
So, in terms of vector it is written as
b⃗=1i^−1j^+1k^\vec b = 1\hat i - 1\hat j + 1\hat k
And the direction ratios of second equation is (1,−3,0)\left( {1, - 3,0} \right)
So, in terms of vector it is written as
c⃗=1i^−3j^+0k^\vec c = 1\hat i - 3\hat j + 0\hat k
Now, it is a known fact that the direction ratiosl,m,nl,m,nis perpendicular to the given equations
Therefore individually multiplication of direction ratios should be zero i.e. its dot product is zero.
i.e. a⃗.b⃗=0, a⃗.c⃗=0\vec a.\vec b = 0,{\text{ }}\vec a.\vec c = 0.

⇒1.l−1.m+1.n=0 ⇒1.l−3.m+0.n=0  \Rightarrow 1.l - 1.m + 1.n = 0 \\\ \Rightarrow 1.l - 3.m + 0.n = 0 \\\
Now, solve these two equations
l((−1×0)−(−3×1))=m1−0=n−3−(−1) l3=m1=n−2  \dfrac{l}{{\left( {\left( { - 1 \times 0} \right) - \left( { - 3 \times 1} \right)} \right)}} = \dfrac{m}{{1 - 0}} = \dfrac{n}{{ - 3 - \left( { - 1} \right)}} \\\ \dfrac{l}{3} = \dfrac{m}{1} = \dfrac{n}{{ - 2}} \\\
Therefore the direction ratios of the line x−y+z−5=0=x−3y−6x - y + z - 5 = 0 = x - 3y - 6are proportional to (3,1,−2)\left( {3,1, - 2} \right)
Hence option (a) is correct.

Note: - In such types of questions the key concept we have to remember is that the direction ratiosl,m,nl,m,nis always perpendicular to the line equations so, individually multiplication of direction ratios should be zero, i.e. its dot product is zero, then simplify these equations we will get the required answer.