Task 243

The task

Tangents are drawn from $(1,3)$ to the parabola $y^2 = 4x,$ at $A$ and $B.$ Find the length $AB.$ [asy] unitsize(0.4 cm); real upperparab (real x) { return (sqrt(4*x)); } real lowerparab (real x) { return (-sqrt(4*x)); } pair A, B, P; P = (1,3); A = ((7 + 3*sqrt(5))/2, upperparab((7 + 3*sqrt(5))/2)); B = ((7 - 3*sqrt(5))/2, upperparab((7 - 3*sqrt(5))/2)); draw(graph(upperparab,0,10)); draw(graph(lowerparab,0,10)); draw(interp(A,P,-0.8)--interp(A,P,1.2)); draw(interp(B,P,-1)--interp(B,P,1.5)); dot("$A$", A, N); dot("$B$", B, W); dot("$(1,3)$", P, NW); [/asy]