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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.
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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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