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Home » O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
CBSE | Class 12 | Exercise 1 | Maths | NCERT Exemplar | Three Dimensional Geometry
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.

According to ques,

    \[~O\text{ }\left( 0,\text{ }0,\text{ }0 \right)\text{ }and\text{ }A\left( a,\text{ }b,\text{ }c \right)\]

Hence,

the direction ratios of OA :

NCERT Exemplar Solutions Class 12 Mathematics Chapter 11 - 20

    \[a\text{ }\text{ }0,\text{ }b\text{ }\text{ }0,\text{ }c\text{ }\text{ }0\text{ }=\text{ }a,\text{ }b,\text{ }c\]

also, the direction ratios of the normal to the plane are

    \[\left( a,\text{ }b,\text{ }c \right).\]

Since, the equation of the plan passing through the point A(a, b, c) is:

    \[a\left( x\text{ }\text{ }a \right)\text{ }+\text{ }b\left( y\text{ }\text{ }b \right)\text{ }+\text{ }c\left( z\text{ }\text{ }c \right)\text{ }=\text{ }0\]

Or,

    \[ax\text{ }\text{ }{{a}^{2}}~+\text{ }by\text{ }\text{ }{{b}^{2}}~+\text{ }cz\text{ }\text{ }{{c}^{2}}~=\text{ }0\]

Or,

    \[ax\text{ }+\text{ }by\text{ }+\text{ }cz\text{ }=\text{ }{{a}^{2}}~+\text{ }{{b}^{2}}~+\text{ }{{c}^{2}}\]

Hence,

Required equation of the plane is:

    \[ax\text{ }+\text{ }by\text{ }+\text{ }cz\text{ }=\text{ }{{a}^{2}}~+\text{ }{{b}^{2}}~+\text{ }{{c}^{2}}\]

 

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