P(→p) and Q(→q) are the position vectors of two fixed points and R(→r) is the position vector of a variable point. If R moves such that (→r−→p)×(→r−→q)=→0, then the locus of R is
A
A plane containing the origin O and parallel to two non-collinear vectors →OP and →OQ
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B
The surface of a sphere described on PQ as its diameter
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C
A line passing through points P and Q
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D
A set of lines parallel to line PQ
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Solution
The correct option is D A line passing through points P and Q (→r−→p)×(→r−→q)=0 →r×(→r−→q)−→p×(→r−→q)=0 →r×→r−→r×→q−→p×→r+→p×→q=0 →p×→q−→r×→q−→p×→r=0 →p×→q−→r×→q+→r×→p=0 →p×→q+→r×(→p−→q)=0 →p×→q=→r×(→q−→p) Therefore →r=→p+t(→q−→p) This represents a line passing through P and Q.