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If A and B be the points \[\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)\]and \[\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)\], respectively, find the equation of the set of points P such that \[\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}\], where k is a constant.
If A and B be the points \[\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)\]and \[\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)\], respectively, find the equation of the set of points P such that \[\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}\], where k is a constant.
Class 11 | Maths | Miscellaneous Exercise | NCERT | Three Dimensional Geometry
If A and B be the points \[\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)\]and \[\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)\], respectively, find the equation of the set of points P such that \[\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}\], where k is a constant.

Given:

The points A \[\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)\]and B \[\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)\]

\[\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}\]……….(1)

Let the point be P (x, y, z).

Now by using distance formula,

We know that the distance between two points \[P\text{ }({{x}_{1}},\text{ }{{y}_{1}},\text{ }{{z}_{1}})\]and \[Q\text{ }({{x}_{2}},\text{ }{{y}_{2}},\text{ }{{z}_{2}})\] is given by

So,

And

Now, substituting these values in (1), we have

\[[{{\left( 3\text{ }\text{ }x \right)}^{2}}~+\text{ }{{\left( 4\text{ }\text{ }y \right)}^{2}}~+\text{ }{{\left( 5\text{ }\text{ }z \right)}^{2}}\left] \text{ }+\text{ } \right[{{\left( -1\text{ }\text{ }x \right)}^{2}}~+\text{ }{{\left( 3\text{ }\text{ }y \right)}^{2}}~+\text{ }{{\left( -7\text{ }\text{ }z \right)}^{2}}]\text{ }=\text{ }{{k}^{2}}\]

\[[(9\text{ }+\text{ }{{x}^{2}}~\text{ }6x)\text{ }+\text{ }(16\text{ }+\text{ }{{y}^{2}}~\text{ }8y)\text{ }+\text{ }(25\text{ }+\text{ }{{z}^{2}}~\text{ }10z)\left] \text{ }+\text{ } \right[(1\text{ }+\text{ }{{x}^{2}}~+\text{ }2x)\text{ }+\text{ }(9\text{ }+\text{ }{{y}^{2}}~\text{ }6y)\text{ }+\text{ }(49\text{ }+\text{ }{{z}^{2}}~+\text{ }14z)]\text{ }=\text{ }{{k}^{2}}\]

\[9\text{ }+\text{ }{{x}^{2}}~\text{ }6x\text{ }+\text{ }16\text{ }+\text{ }{{y}^{2}}~\text{ }8y\text{ }+\text{ }25\text{ }+\text{ }{{z}^{2}}~\text{ }10z\text{ }+\text{ }1\text{ }+\text{ }{{x}^{2}}~+\text{ }2x\text{ }+\text{ }9\text{ }+\text{ }{{y}^{2}}~\text{ }6y\text{ }+\text{ }49\text{ }+\text{ }{{z}^{2}}~+\text{ }14z\text{ }=\text{ }{{k}^{2}}\]

\[2{{x}^{2}}~+\text{ }2{{y}^{2}}~+\text{ }2{{z}^{2}}~\text{ }4x\text{ }\text{ }14y\text{ }+\text{ }4z\text{ }+\text{ }109\text{ }=\text{ }{{k}^{2}}\]

\[2{{x}^{2}}~+\text{ }2{{y}^{2}}~+\text{ }2{{z}^{2}}~\text{ }4x\text{ }\text{ }14y\text{ }+\text{ }4z\text{ }=\text{ }{{k}^{2}}~\text{ }109\]

\[2\text{ }({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }{{z}^{2}}~\text{ }2x\text{ }\text{ }7y\text{ }+\text{ }2z)\text{ }=\text{ }{{k}^{2}}~\text{ }109\]

\[({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }{{z}^{2}}~\text{ }2x\text{ }\text{ }7y\text{ }+\text{ }2z)\text{ }=\text{ }({{k}^{2}}~\text{ }109)/2\]

Hence, the required equation is \[({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }{{z}^{2}}~\text{ }2x\text{ }\text{ }7y\text{ }+\text{ }2z)\text{ }=\text{ }({{k}^{2}}~\text{ }109)/2\]

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