Question

If an iron rod of length $1$ unit is always moving with its ends in contact with coordinate axes and the locus of mid point of the rod is $x^2+y^2=k$ then the value of $16k$ is ​​​​​​​

Answer 1

Let $(x,y)$ is the mid-point of rod .
End points are $A(2x,0)$ and $B(0,2y)$
Given $AB=1$

$\Rightarrow 4x^2+4y^2=1$
$\Rightarrow x^2+y^2=\frac{1}{4}$
So, the locus of mid point of the rod is $x^2+y^2=\frac{1}{4}$
On comparing with the given equation, we get
$k=\frac{1}{4}\Rightarrow 16k=16\times \frac{1}{4}=4$