Let O be the origin and OP and OQ be the tangents to the circle x ^...

The given information can be represented in the form of the diagram below.

Since angle in a semicircle is a right angle.

Hence, the other circle will be passing through the centre of the first circle.

The two ends of diameter of the circle is $O\left(0, 0\right)\text{and}C\left(3, - 2\right).</p><p>Hence,$ the $required equation of circle is \left(x - 0\right)$\left(x - 3\right) + \left(y - 0\right) \left(y - \left(- 2\right)\right) = 0$</p><p>$⇒ {x}^{2} + {y}^{2} - 3x + 2y = 0$</p><p>Put \left(\alpha, \dfrac{1}{2}\right)$ in the above equation$⇒ {\alpha}^{2} + \dfrac{1}{4} -3\alpha + 1 = 0$
$⇒ {\alpha}^{2} - 3\alpha + \dfrac{5}{4} =0$
$⇒ 4{\alpha}^{2} - 12\alpha + 5 = 0$

$⇒ 4{\alpha}^{2} - 10\alpha - 2\alpha + 5 = 0$

$⇒ 2\alpha ( 2\alpha - 5 ) - 1 ( 2\alpha - 5 ) = 0⇒ \alpha = \dfrac{1}{2}, \dfrac{5}{2}$

Therefore this is$the correct option.$