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  • Please Solve RD Sharma Class 12 Chapter 28 The Plane Exercise 28.11 Question 10 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 28 The Plane Exercise 28.11 Question 10 Maths Textbook Solution.

Answers (1)

Answer: The line of section is parallel to plane

Hint: Use properties of plane

Given:

\begin{aligned} &5 x+2 y-4 z+2=0,2 x+8 y+2 z-1=0 \\\\ &\& 4 x-2 y-5 z-2=0 \end{aligned}

Solution: Let  a_{1}, b_{1} \& c_{1}  be the direction ratios of line   5 x+2 y-4 z+2=0,2 x+8 y+2 z-1=0

As we know, that if two planes are perpendicular with direction ratios as a_{1}, b_{1} \& c_{1}  and   a_{2}, b_{2} \& c_{2}   then  a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0

Since line lies in both the planes, so it is perpendicular to both planes

            \begin{aligned} &5 a_{1}+2 b_{1}-4 c_{1}=0 \\ & \end{aligned}                              …………….. (1)

            2 a_{1}+8 b_{1}+2 c_{1}=0                              …………….. (2)

Solving equation (1) and (2) by cross multiplication

We have,

            \begin{aligned} &\frac{a_{1}}{2 \times 2-(-4) \times 8}=\frac{b_{1}}{2 \times(-4)-5 \times 2}=\frac{c_{1}}{5 \times 8-2 \times 2} \\ & \end{aligned}

           \Rightarrow \frac{a_{1}}{4+32}=\frac{b_{1}}{-8-10}=\frac{c_{1}}{40-4}

           \begin{aligned} &\Rightarrow \frac{a_{1}}{36}=\frac{b_{1}}{-18}=\frac{c_{1}}{36}=k \\ & \end{aligned}

           \Rightarrow \frac{a_{1}}{2}=\frac{b_{1}}{-1}=\frac{c_{1}}{2}=k \\

           a=2 k, b=-k, c=2 k

We know that line

           \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}}   is parallel to plane  a_{2} x+b_{2} y+c_{2} z+d_{2}=0

if   a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0            ………….. (3)

Here, line with direction ratios   is parallel to plane

            \begin{aligned} &4 x-2 y-5 z-2=0 \\ & \end{aligned}

            \Rightarrow 2 \times 4+(-1) \times(-2)+2 \times(-5)=0 \\

            \Rightarrow 8+2-10=0

Therefore, the line of section is parallel to the plane.

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