The circle passing through the point $(-1, 0)$ and touching the $y$-axis at $(0, 2)$ also passes through the point.

$\text{Given: circle passes through } (-1, 0)$

$\text{and touches the } y\text{-axis at } (0, 2)$

$\text{To find: which point also lies on the circle}$

$A(-1, 0), \quad T(0, 2)$

$\text{At } T:\ \text{radius} \perp \text{tangent (the } y\text{-axis)}$

$\Rightarrow \text{centre } O = (h, 2), \quad r = |h|$

$OA^2 = (h+1)^2 + (2-0)^2 = r^2 = h^2$

$h^2 + 2h + 5 = h^2 \ \Rightarrow \ 2h + 5 = 0$

$h = -\dfrac{5}{2}$

$O\left(-\dfrac{5}{2}, 2\right), \quad r = \dfrac{5}{2}$

$\left(x + \dfrac{5}{2}\right)^2 + (y - 2)^2 = \dfrac{25}{4}$

$\left(x + \tfrac52\right)^2 + (y-2)^2 = \tfrac{25}{4} \ \Rightarrow \ x^2 + y^2 + 5x - 4y + 4 = 0$

hline Point x²+y²+5x-4y+4 On circle? hline (-3/2,0) -5/4 × (-5/2,2) -25/4 × (-3/2,5/2) -5 × (-4,0) 0 checkmark hline

$\boxed{\text{Answer: } (-4, 0) \ \text{— option (4)}}$