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TS EAMCET · Maths · Three Dimensional Geometry

Consider the following statements: Assertion (A): The direction ratios of a line \(\mathrm{L}_1\) are \(2,5,7\) and the direction ratios of another line \(\mathrm{L}_2\) are \(\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}\). Then the lines \(\mathrm{L}_1, \mathrm{~L}_2\) 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 \(\mathrm{L}_2\) are \(\mathrm{a}_2, \mathrm{~b}_2, \mathrm{c}_2\) and \(\mathrm{a}_1 \mathrm{a}_2+\mathrm{b}_1 \mathrm{~b}_2\) \(+c_1 c_2=0\), then the lines of \(L_1, L_2\) are parallel Which one of the following is True?

  1. A (A) and (R) are true. (R) is the correct explanation of (A)
  2. B (A) and (R) are true, but (R) is not the correct explanation of \((\mathrm{A})\)
  3. C (A) is true, (R) is false
  4. D (A) is false, (R) is true
Verified Solution

Answer & Solution

Correct Answer

(C) (A) is true, (R) is false

Step-by-step Solution

Detailed explanation

D.R. of \(L_1:\left(a_1, b_1, c_1\right)\) D.R. of \(L_2:\left(a_2, b_2, c_2\right)\) If \(L_1 \| L_2 \Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) \(\therefore(R)\) is false.