d _ 1 and d _ 2 are the impact parameters corresponding to...

The relation between impact parameter and scattering angle is $b = \dfrac{kz{e}^{2}}{E}\cot \left(\dfrac{\theta}{2}\right)$

$⇒ b ∝ \cot \left(\dfrac{\theta}{2}\right)$

$⇒ \sqrt{{d}_{1}} ∝ \cot \left(\dfrac{60 °}{2}\right)$

$⇒ \sqrt{{d}_{2}} ∝ \cot \left(\dfrac{90 °}{2}\right)$

$\dfrac{\sqrt{{d}_{1}}}{\sqrt{{d}_{2}}} = \dfrac{\cot \left(\dfrac{60 °}{2}\right)}{\cot \left(\dfrac{90 °}{2}\right)} = \sqrt{3} ⇒ {d}_{1} = 3{d}_{2} ⇒ x = 3$