A parabola has its focus on the positive X-axis and the Y-axis as its directrix. If P( , 4) is a point on this parabola such that the tangent to the parabola at point P passes through the origin, then the distance of P from origin is

Step-by-step solution

Let the focus of the parabola be F(a, 0) with a > 0 . Since the directrix is the Y-axis ( x = 0 ), the distance of any point (x, y) on the parabola from the focus is equal to its distance from the directrix. (x - a)^2 + y^2 = x Squaring both sides, we get: (x - a)^2 + y^2 = x^2 x^2 - 2ax + a^2 + y^2 = x^2 y^2 = 2ax - a^2 The point P( , 4) lies on the parabola, so: 16 = 2a - a^2 The equation of the tangent to the parabola y^2 = 2ax - a^2 at P( , 4) is given by T = 0 : 4y = a(x + ) - a^2 Since the tangent passes thro