Semi latus rectum of the parabola $y^{2} = 4 a x$ , is the _____ mean between segments of any focal chord of the parabola.

Arithmetic

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Geometric

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Harmonic

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None of these

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Solution

The correct option is C
Harmonic

Standard parabola

$y^{2} = 4 a x$ is given

Length of latus rectum =4a

Semi latus rectum=2a

Let the coordinates of P be $(a t_{1}^{2}, 2 a t_{1})) a n d Q ((a t_{2}^{2}, 2 a t_{2})$

$P S = \sqrt{(a - a t_{1}^{2})^{2} + (2 a t_{1} - 0)^{2}} = a (1 - t_{1}^{2})^{2} + 4 a t_{1}^{2}$

$P S = \sqrt{a^{2} (1 + t_{1}^{2})^{2}} = a (1 + t_{1}^{2})$

$S Q = \sqrt{(a - a t_{2}^{2})^{2} + (2 a t_{2} - 0)^{2}} = \sqrt{a (1 - t_{2}^{2})^{2} + 4 a t_{2}^{2}}$

$= \sqrt{a^{2} (1 + t_{2}^{2})^{2}} = a (1 + t_{2}^{2})$

Segment of the focul chord SP $&$ SQ.

We need tha AM,GM and HM of these segments

Let's calculate HM

$H M = \frac{2 (S P) (S Q)}{S P + S Q} = \frac{2 a (1 + t_{1}^{2}) \times a (1 + t_{2}^{2})}{a (1 + t_{1}^{2}) + a (1 + t_{2}^{2})}$

$= \frac{2 a^{2} (1 + t_{1}^{2}) (1 + t_{2}^{2})}{a (1 + t_{1}^{2} + 1 + t_{2}^{2})}$

$= \frac{2 a (1 + t_{1}^{2}) (1 + t_{2}^{2})}{2 + t_{1}^{2} + t_{2}^{2})}$ ............(1)

Here, equation of focal chord

$y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$

$y = 2 a t_{1} = \frac{2 a t_{2} - 2 a t_{1}}{a t_{2}^{2} - a t_{1}^{2}} (x - a t_{1}^{2})$

this passes through focus (a,0)

$0 - 2 a t_{1} = \frac{2 (t_{2} - t_{1})}{(t_{2} - t_{1}) (t_{2} + t_{1})} (a - a t_{1}^{2})$

$- 2 a t_{1} = \frac{2}{t_{1} + t_{2}} a (1 - t_{1}^{2})$

$- t_{1} = \frac{1 - t_{1}^{2}}{t_{1} + t_{2}}$

$- t_{1}^{2} - t_{1} . t_{2} = 1 - t_{1}^{2}$

$t_{2} = \frac{- 1}{t_{1}}$

Substituting $t_{2} = \frac{- 1}{t_{1}}$ in equation (1)

$H M = \frac{2 a (1 + t_{1}^{2}) (1 + \frac{1}{t_{1}^{2}})}{2 + t_{1}^{2} + \frac{1}{t_{1}^{2}}}$

$= \frac{\frac{2 a (1 + t_{1}^{2}) (1 + t_{1}^{2})}{t_{1}^{2}}}{\frac{2 t_{1}^{2} + t_{1}^{4} + 1}{t_{1}^{2}}}$

$= \frac{2 a (1 + t_{1}^{2})^{2}}{(t_{1}^{2} + 1)^{2}} = 2 a =$ Semi latus rectum

So, semi latus rectum of the parabola $y^{2} = 4 a x$ is the harmonic

mean between segments of any focal chord of parabola.

Similarly, we can try for AM $&$ GM. We will find that it won't satisfy

the given condition of the question.

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