question_answer

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

A)  \[{{x}^{2}}+{{y}^{2}}=4\]   

B)  \[{{x}^{2}}+{{y}^{2}}=16\]

C)  \[{{x}^{2}}+{{y}^{2}}=8\]    

D)  \[|x|+|y|=8\]

Correct Answer: B

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 \[{{x}^{2}}+{{y}^{2}}=16\]