Let C be the circle with center at (1, 1) and radius = 1 . If T is...

Let $r$ be the required radius
As origin lies on the circle, the distance of $\left(0, 0\right)$ from \left(0, y\right) = r$</p><p>$∴ \text{y} = \text{r}$<br /><br /><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/ca900165-d006-4374-a67d-bcde7d10e525-image.png" style="width: 230px; height: 187px;" /><br /><br />As the two circles touch externally, distance between centres is sum of the radii<br /><br />$⇒ \sqrt{{\left(0 - 1\right)}^{2} + {\left(y - 1\right)}^{2}} = r + 1$<br /><br />$⇒ 1 + {y}^{2} - 2y + 1 = {r}^{2} + 2r + 1$<br /><br />$⇒ 1 + {r}^{2} - 2r = {r}^{2} + 2r$, \left[∵ y = r\right]$

$∴ 1 = 4r$

$∴ r = \dfrac{1}{4}$