Question
If the sum of the distances of a point from two perpendicular lines in a plane is $$1,$$ then its locus is :
A.
square
B.
circle
C.
straight line
D.
two intersecting lines
Answer :
square
Solution :
Let the two perpendicular lines be the coordinate axes and let the point be $$P\left( {h,\,k} \right).$$
Then sum of the distances of $$P\left( {h,\,k} \right)$$ from the coordinate axes is $$\left| h \right| + \left| k \right|.$$ It is given that $$\left| h \right| + \left| k \right| = 1.$$
So, locus of $$\left( {h,\,k} \right)$$ is $$\left| x \right| + \left| y \right| = 1.$$ This gives four lines $$x + y = 1,\, x - y = 1,\, -x + y = 1,\, -x - y = 1$$ which enclose a square.