A(1,0) is a point on the parabola y=2x(x−1). From point A, point P is moving along the curve towards the origin O(0,0). As P → O, sec^2∠APO → N, where N is a positive integer. What is the value of N?

Question

A(1,0) is a point on the parabola y=2x(x−1). From point A, point P is moving along the curve towards the origin O(0,0). As P → O, sec^2∠APO → N, where N is a positive integer. What is the value of N?

Answer 1

Let θ = ∠POA, φ = ∠PAO, β=APO

tanθ = y/x = 2x(x-1)/x = 2(x-1)
So, as P->O, tanθ -> -2
So, cosidering only |tanθ|, we have
θ = arctan(2)

Now, θ+φ+β = π, so since φ->0,
β = π-θ-φ = π - arctan(2)

cosβ -> cos(π - arctan(2)) = 1/√5
so, sec^2 β -> 5