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  • Please Solve RD Sharma Class 12 Chapter 28 The Plane Exercise 28.11 Question 22 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 28 The Plane Exercise 28.11 Question 22 Maths Textbook Solution.

Answers (1)

Answer:  \sin ^{-1}\left(\frac{8}{21}\right)

Hint: Use formula  \sin \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}

Given:  \frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6} \text { and } 10 x+2 y-11 z=3

Solution: Direction ratios of the line

            \frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6} \text { are }(2,3,6)

Direction ratio of a line perpendicular to the plane  10 x+2 y-11 z=3 \text { are }(10,2,-11)

As we know that the angle \theta  between the line \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}}  and plane

a_{2} x+b_{2} y+c_{2} z+d_{2}=0  is given by

            \begin{aligned} &\sin \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} \\ & \end{aligned}

            \sin \theta=\frac{2 \times 10+3 \times 2+6 \times(-11)}{\sqrt{2^{2}+3^{2}+6^{2}} \sqrt{10^{2}+2^{2}+(-11)^{2}}}

            \begin{aligned} \sin \theta &=\frac{-40}{\sqrt{49} \sqrt{225}} \\ & \end{aligned}

                     =\frac{-40}{7 \times 15}=\frac{-8}{21}

           \begin{gathered} \sin \theta=\frac{-8}{21} \\ \end{gathered}

                \theta=\sin ^{-1}\left(\frac{8}{21}\right)

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