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Line and Plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane2 Marks
MCQ
The acute angle between the line joining the points $(2,1,-3),(-3,1,7)$ and a line parallel to $\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}$ through the point $(-1,0,4)$ is
  • ✓
    $\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)$
  • B
    $\cos ^{-1}\left(\frac{1}{\sqrt{10}}\right)$
  • C
    $\cos ^{-1}\left(\frac{3}{5 \sqrt{10}}\right)$
  • D
    $\cos ^{-1}\left(\frac{1}{5 \sqrt{10}}\right)$
✓

Answer

Correct option: A.
$\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)$
(A)
The d.r.s. of the line joining the points $(2,1$, $-3)$ and $(-3,1,7)$ are $-5,0,10$
The d.r.s. of the line parallel to line
$\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}$ are $3,4,5$
$\therefore \quad$ The angle between the lines having d.r.s. $-5,0,10$ and $3,4,5$ is
$\cos \theta=\left|\frac{-5(3)+0(4)+10(5)}{\sqrt{25+0+100} \sqrt{9+16+25}}\right|$
$\Rightarrow \cos \theta=\frac{35}{25 \sqrt{10}}$
$\Rightarrow \theta=\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)$

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