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I was trying to solve one question which is asking to find a plane which passes through given point and is parallel to given line.

The given point is $M(2,-5,3)$ and the given line is given as an interesection of the planes $2x-y+3z-1=0 \text{ and } 5x+4y-z-7=0$

It is still unclear for me why there is only one unique plane which can be answer, I think that there are more possible planes that can be answers to this.

someone123123
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3 Answers3

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Your are right, such plane is not unique. For example the planes $2x-y+3z=18$ and $5x+4y-z=-13$ pass through the point $(2,-5,3)$ and they are parallel to the given line.

More generally, through the given point, there is a unique line parallel to the given line, but then any plane through this second line is parallel to the given line.

Robert Z
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  • I'm thinking like this, but my teacher gave some explanation that I didn't really understand. She says that there is only one such plane – someone123123 Mar 13 '19 at 09:57
  • Maybe the problem was "find a plane which passes through a given point and it is orthogonal to a given line. – Robert Z Mar 13 '19 at 10:00
  • Or perhaps the plane is supposed to include the intersection line of the two given planes (see updated question). Impossible to know without seeing the teacher’s solution. – amd Mar 14 '19 at 00:24
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You are right: there are infinitely mane planes passing through a point and parallel to a given line.

Alessio Del Vigna
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As another answer points out, the claim is false. Given a set of planes parallel to each other as well as to the given line, only one of those planes will pass through the given point.

Oscar Lanzi
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