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  • Please solve RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 9 maths textbook solution

Please solve RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 9 maths textbook solution

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Final Answer:   5\sqrt{2}

Hint:

Length of projection =  Length ×  Direction cosine

Given:

Projections on coordinate axes are 3,4,5

To Find: 

Length of line segment

Solution:

Let Line segment PQ makes angle α,β,γ with x,y and z axes, such that length of projection in coordinate axes are:

\begin{aligned} &P Q \cos \alpha=3 \Rightarrow \cos \alpha=\frac{3}{P Q} \\ &P Q \cos \beta=4 \Rightarrow \cos \beta=\frac{4}{P Q} \\ &P Q \cos \gamma=5 \Rightarrow \cos \gamma=\frac{5}{P Q} \end{aligned}

Now by properties of direction cosine

\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1

\begin{aligned} &\therefore\left(\frac{3}{P Q}\right)^{2}+\left(\frac{4}{P Q}\right)^{2}+\left(\frac{5}{P Q}\right)^{2}=1 \\\\ &\frac{9}{P Q^{2}}+\frac{16}{P Q^{2}}+\frac{25}{P Q^{2}}=1 \\\\ &\frac{50}{P Q^{2}}=1 \end{aligned}

\begin{aligned} &P Q^{2}=50 \\\\ &P Q=\sqrt{50} \\\\ &P Q=5 \sqrt{2} \end{aligned}

Therefore, the total length of line segment is 5\sqrt{2}

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