Question

If \(P\) is any point on the curve \[ y^2=4ax, \] other than the origin, then the length of the subtangent at \(P\), \(y\)-coordinate of \(P\) and the length of the subnormal at \(P\) are in

• A. arithmetic progression
• B. arithmetic-geometric progression
• C. harmonic progression
• D. geometric progression

Hint:

For the parabola \[ \boxed{y^2=4ax}, \] at the point \[ (at^2,2at), \] \[ \boxed{\text{Subtangent}=2at^2,\qquad \text{Subnormal}=2a.} \] Compare these with the \(y\)-coordinate to identify the progression.

Answer 1

The Correct Option is D

Step 1: Take the parametric coordinates of the parabola. For the parabola \[ y^2=4ax, \] a general point is \[ P(at^2,\;2at). \] Hence, \[ y=2at. \]

Step 2: Find the subtangent and subnormal. The slope of the tangent is \[ \frac{dy}{dx} = \frac1t. \] Therefore, Length of subtangent: \[ y\cdot\frac{dx}{dy} = y\left(\frac1{dy/dx}\right) = 2at\cdot t = 2at^2. \] Length of subnormal: \[ y\cdot\frac{dy}{dx} = 2at\cdot\frac1t = 2a. \] Thus, the three quantities are \[ 2at^2,\qquad 2at,\qquad 2a. \]

Step 3: Check the progression. Now, \[ (2at)^2 = (2at^2)(2a). \] Hence, \[ (\text{\(y\)-coordinate})^2 = (\text{subtangent}) \times (\text{subnormal}). \] Therefore, the three quantities are in \[ \boxed{\text{geometric progression}.} \] Thus, \[ \boxed{(D)} \] is the correct answer.