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

Answer: 4

$|OP|=|OQ|=a$
$|OM|=\frac{|0-0-a|}{\sqrt{1^2+(-1{)}^2}}$
$=\frac{a}{\sqrt{2}}$
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 $|a|=(0,4)$
$\Rightarrow k=4$
$\Rightarrow [k]=4$