Question

If a chord of the circle $x^{2}+y^{2}=8$ makes equal intercepts of length $a$ on the coordinate axes and the range of values of $|a|$ is $(0, k)$, then find $[k]$, where $[ \,\,\,] $represents the greatest integer function.

Answer 1

$| OP | =| OQ |=a $
$| OM | =\frac{|0-0-a|}{\sqrt{1^{2}+(-1)^{2}}}$
$=\frac{a}{\sqrt{2}}$
Also, $| OM |=$ radius of the circle $x^{2}+y^{2}=8$
$=2 \sqrt{2}$
For limiting case,
$\Rightarrow \frac{a}{\sqrt{2}}=2 \sqrt{2} $
$\Rightarrow a=4$
(In limiting case, the chord tends to become the tangent $PQ$.)
$\Rightarrow$ The range of values of $|a|=(0,4)$
$\Rightarrow k=4$
$ \Rightarrow[k]=4$