Assertion (A) : The direction ratios of a line $L_{1}$ are 2,5,7 and the direction ratios of another line $L_{2}$ are $\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}} .$ Then the lines $L_{1}, L_{2}$ are parallel
Reason (R) : If the direction ratios of a line $L_{1}$ are $a_{1}, b_{1}, c_{1},$ the direction ratios of a line $L_{2}$ are $a_{2}, b_{2}, c_{2}$ and $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ then the lines of $L_{1}, L_{2}$ are parallel
Which one of the following is True?

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(A) and (R) are true. (R) is the correct explanation of (A)

(A) and (R) are true, but (R) is not the correct explanation of (A)

(A) is false, (R) is true

(A) is true, (R) is false

answer is C.

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Detailed Solution

$(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2}) - {(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2})}^{2} = 0$

${(a_{1} b_{2} - a_{2} b_{1})}^{2} + {(b_{1} c_{2} - b_{2} c_{1})}^{2} + {(c_{1} a_{2} - c_{2} a_{1})}^{2} = 0 \Rightarrow a_{1} b_{2} = a_{2} b_{1}, b_{1} c_{2} = b_{2}$

$c_{1}, c_{1} a_{2} = c_{2} a_{1} ∴ \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ ,lines are parallel

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