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I was solving some objective problems on locus based problems and then I encountered this, If the sum of the distances of a point $P$ from two perpendicular lines in a plane is $1$,then find the locus of the point $P$.

I am not able to figure out the exact picture but what I think that it will be a circle. Please help me. Thanks

Anish Ray
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  • If we think of the perpendicular lines as $x$-axis and $y$-axis, then the locus would be $|x|+|y|=1$ – Lozenges Jun 02 '17 at 13:00

2 Answers2

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The locus of points a fixed distance away from a given line is 2 lines, parallel to the first line and to each other. For two lines, you have to find the intersection of their respective loci. When you draw these out, you will see the 4 lines create a square, with side 2, and the 4 vertices of the square are the locus you are looking for.

Paul
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I simplify the case by taking the two perpendicular lines to be the x- and the y-axes.enter image description here

In the picture, there is the line $L_1 : x + y = 1$. In which, A(0, 1), B (0.4, 0.6), C, D, E, F, H and K are some of the sample points on it. Clearly, only those points marked in red (and also those lying between A and F) meet the requirement but H, K are too far away. Therefore, part of the required locus is the line segment AF.

Following the same reasoning, the other three green dotted line segments also meet the requirement.

Mick
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