Q213 / 270 · NDA 2023 (Phase 1) · Mathematics

Paragraph: Consider two lines whose direction ratios are $(2,-1,2)$ and $(k, 3,5)$. They are inclined at an angle $\frac{\pi}{4}$.
Question: What is the value of $k$ ?

  • Correct answer
    A
    $4$
  • B
    $2$
  • C
    $1$
  • D
    $-1$

Solution

$A(2,-1,2) B(k, 3,5)$ $\cos \theta=\frac{2 k-3+10}{\sqrt{2^2+(-1)^2+2^2} \cdot \sqrt{k^2+3^2+5^2}}$ $\Rightarrow \quad \cos \frac{\pi}{4}=\frac{2 k+7}{3 \cdot \sqrt{34+k^2}}=\frac{1}{\sqrt{2}}$ $\Rightarrow \quad 3 \sqrt{34+k^2}=\sqrt{2}(2 k+7)$ $\Rightarrow \quad 9\left(34+k^2\right)=2(2 k+7)^2$ $\Rightarrow \quad 306+9 k^2=8 k^2+98+56 k$ $\Rightarrow \quad k^2-56 k+208=0$ $\Rightarrow \quad(k-52)(k-4)=0$ $\Rightarrow \quad k=4,52$ $\therefore \quad k=4$

Hence, the correct answer is option: 'A'.

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