Question

A rod of length I sides with its ends on two perpendicular lines. Then, the locus of its midpoint is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Let the two perpendicular lines be the coordinate axes. let AB be rod of lenght/ and the coordinates of A and B be(a,0) and (0,b) respectively.

Let P(h,k) be the mid point of the rod AB in one of the infinite position it attains, then

$h=\frac{a+0}{2}andk=\frac{0+b}{2}\Rightarrow h=\frac{a}{2}andk=\frac{b}{2}\cdots \cdots \cdots \left(i\right)From\Delta OAB,wehaveA{B}^2=O{A}^2+O{B}^2\Rightarrow a^2+b^2=I^2\Rightarrow (2h{)}^2+(2k{)}^2=I^2\Rightarrow 4h^2+4k^2=I^2\Rightarrow h^2+x^2=\frac{I^2}{4}\therefore Theequationoflocusis{x}^2+y^2=\frac{I^2}{4}Hence,\left(a\right)isthecorrectanswer.$