MHT CET · Maths · Three Dimensional Geometry
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
- A \(\cos ^{-1}\left(\frac{1}{\sqrt{10}}\right)\)
- B \(\cos ^{-1}\left(\frac{5}{7 \sqrt{10}}\right)\)
- C \(\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)\)
- D \(\cos ^{-1}\left(\frac{3}{5 \sqrt{10}}\right)\)
Answer & Solution
Correct Answer
(C) \(\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)\)
Step-by-step Solution
Detailed explanation
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
\(\cos \theta=\left|\frac{-5(3)+0(4)+10(5)}{\sqrt{25+0+100} \sqrt{9+16+25}}\right|\)
\(\begin{aligned} & \Rightarrow \cos \theta=\frac{35}{25 \sqrt{10}} \\ & \Rightarrow \theta=\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)\end{aligned}\)
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
\(\cos \theta=\left|\frac{-5(3)+0(4)+10(5)}{\sqrt{25+0+100} \sqrt{9+16+25}}\right|\)
\(\begin{aligned} & \Rightarrow \cos \theta=\frac{35}{25 \sqrt{10}} \\ & \Rightarrow \theta=\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)\end{aligned}\)
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