Q97 / 160 · AP EAMCET 2022 (05 Jul Shift 1) · Mathematics

If the direction cosines of a line are $\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$ and $c-a=4$, then $c a=$

  • A
    $24$
  • Correct answer
    B
    $21$
  • C
    $18$
  • D
    $33$

Solution

Direction cosine of $a \hat{i}+b \hat{j}+c \hat{k}$ can be $\frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}}$ According to question, $\sqrt{a^2+b^2+c^2}=\sqrt{83}, b=5$ $\Rightarrow \quad a^2+(5)^2+c^2=83$ $\Rightarrow \quad a^2+c^2=83-25$ $\Rightarrow a^2+c^2=58$ ...(i) and $c-a=4$ [given] $(c-a)^2+2 a c=a^2+c^2$ $\Rightarrow(4)^2+2 a c=58 \quad$ [from Eq. (i), $\left.c^2+a^2=58\right]$ $\Rightarrow 2 a c=58-16$ $a c=\frac{42}{2}=21$

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

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