Q. A line segment of 8 units in length moves so that its end points are always on the coordinate axes. Then, the equation of locus of its mid-point is
J & K CETJ & K CET 2011
Solution:
Let the line segment whose length is 8 unit intercept the coordinate axes at
$ A(a,0) $ and $ B(0,b) $
x-axes and y-axes respectively.
Let $ P(h,k) $ be the mid-point of line segment AB.
$ \Rightarrow $ $ h=a/2 $ and $ k=b/2 $
$ \Rightarrow $ $ a=2\,h $ and $ b=2k $ ..(i)
Now, $ AB=\sqrt{{{(a-0)}^{2}}+{{(b-0)}^{2}}} $
$ A{{B}^{2}}={{a}^{2}}+{{b}^{2}} $
$ \Rightarrow $ $ {{(8)}^{2}}={{(2h)}^{2}}+{{(2k)}^{2}} $
(from Eq. (i)] $ \Rightarrow $ $ {{h}^{2}}+{{k}^{2}}=16 $
$ \therefore $ The leocus of mid-point P is
$ {{x}^{2}}+{{y}^{2}}=16 $
