A chord is drawn from a point $P(1, t)$ to the parabola $y^2 = 4ax$ which cuts the parabola at A and B. If $PA.PB = 3|t|$, then find the range of t.

Question

A chord is drawn from a point $P(1, t)$ to the parabola $y^2 = 4ax$ which cuts the parabola at A and B. If $PA.PB = 3|t|$, then find the range of t.

I was able to solve the question using parametric form of a line but wasn't able to figure out how to solve it geometrically using properties of parabola and similarity of triangles(if possible).