Four distinct points ( 2k, 3k ) , ( 1, 0 ) , ( 0, 1 ) and ( 0, 0 )...

We know that, $( 1, 0 ) , ( 0, 1 ) , ( 0, 0 )$ lies on $X -$axis, $Y -$axis and on the origin respectively.

So, $( 2k, 3k )$ will lie on circle whose diameter is $AB$.

Equation of the circle will be given by,

$⇒ ( x - 1 ) ( x ) + ( y - 1 ) ( y ) = 0$

$⇒ {x}^{2} + {y}^{2} - x - y = 0 . . . \left(i\right)</p><p>Using$( 2k, 3k ) in $equation \left(i\right),</p><p>$⇒ ( 2k{)}^{2} + ( 3k{)}^{2} - 2k - 3k = 0$</p><p>$⇒ 13{k}^{2} - 5k = 0$</p><p>$⇒ k = 0, k = $\dfrac{5}{13}</p><p>Hence,$k = $\dfrac{5}{13}.</p>$