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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.

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Question

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.

Sum
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Solution

We have A(a, b, c) and O(0, 0, 0)

∴ Direction ratios of OA = a – 0, b – 0, c – 0

∴ Direction cosines of line OA = `"a"/sqrt("a"^2 + "b"^2 + "c"^2)`

`"b"/sqrt("a"^2 + "b"^2 + "c"^2)`

`"c"/sqrt("a"^2 + "b"^2 + "c"^2)`

Now direction ratios of the normal to the plane are (a, b, c).

∴ Equation of the plane passing through the point A(a, b, c) is a(x – a) + b(y – b) + c(z – c) = 0

⇒ ax – a2 + by – b2 + cz – c2 = 0

⇒ ax + by + cz = a2 + b2 + c2

Hence, the required equation is ax + by + cz = a2 + b2 + c2.

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Chapter 11: Three Dimensional Geometry - Exercise [Page 236]

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NCERT Exemplar Mathematics Exemplar [English] Class 12
Chapter 11 Three Dimensional Geometry
Exercise | Q 14 | Page 236

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