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Three Dimensional Geometry

Question

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If the product of distances of the point (1,1,1) from the origin and the plane x – y + z + k = 0 be 5, then find the value of k.

Solution

Let d₁ , d₂ be the distances of the point (1, 1, 1) from the origin (0, 0, 0) and plane x – y + z + k = 0.

therefore space space space space space space space straight d subscript 1 space equals space square root of left parenthesis 1 minus 0 right parenthesis squared plus left parenthesis 1 minus 0 right parenthesis squared plus left parenthesis 1 minus 0 right parenthesis squared end root space equals space square root of 1 plus 1 plus 1 end root space equals space square root of 3 and space space space straight d subscript 2 space equals space fraction numerator open vertical bar 1 minus 1 plus 1 plus straight k close vertical bar over denominator square root of 1 plus 1 plus 1 end root end fraction space equals space fraction numerator open vertical bar 1 plus straight k close vertical bar over denominator square root of 3 end fraction

From the given condition,

straight d subscript 1 space straight d subscript 2 space equals space 5

therefore space space space space square root of 3 space cross times fraction numerator open vertical bar 1 plus straight k close vertical bar over denominator square root of 3 end fraction space equals space 5 space space space space space space rightwards double arrow space space space space open vertical bar 1 plus straight k close vertical bar space equals space 5 space space space space space rightwards double arrow space space space space 1 plus straight k space equals space plus-or-minus 5 therefore space space space space space space straight k space equals space minus 6 comma space space 4.

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

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