Question

A parabola has its focus on the positive X-axis and the Y-axis as its directrix. If \(P(α,4)\) 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

• A. \(4\)
• B. \(\sqrt{20}\)
• C. \(5\)
• D. \(\sqrt{32}\)

Hint:

Build the parabola with the y-axis as directrix, then use the tangent-through-origin condition.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A parabola with focus \((a, 0)\) on the positive X-axis and directrix the Y-axis (the line \(x = 0\)) has its vertex midway at \(\left(\dfrac{a}{2}, 0\right)\), and its axis is the X-axis.

Step 2: Key Formula or Approach:
A point is on the parabola if its distance from the focus equals its distance from the directrix:
\[ \sqrt{(x-a)^2 + y^2} = x \Rightarrow y^2 = 2ax - a^2 \]
The tangent at \((x_1, y_1)\) is \(y y_1 = a(x + x_1) - a^2\).

Step 3: Detailed Explanation:
The tangent at \(P(\alpha, 4)\) passes through the origin. Substitute \((0,0)\):
\[ 0 = a(\alpha) - a^2 \Rightarrow \alpha = a \quad (a \neq 0) \]
Now P lies on the parabola:
\[ 16 = 2a\alpha - a^2 = 2a^2 - a^2 = a^2 \Rightarrow a = 4 \]
So \(\alpha = 4\) and \(P = (4, 4)\).
Distance from the origin:
\[ OP = \sqrt{4^2 + 4^2} = \sqrt{32} \]
Option (A) 4 and (C) 5 are not distances to the point \((4,4)\), and option (B) \(\sqrt{20}\) would correspond to \((2,4)\).

Final Answer:
The distance of P from the origin is \(\sqrt{32}\), option (D). \[ \boxed{\sqrt{32} \text{ (D)}} \]