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  • In any parabola the harmonic mean of intercepts of any focal chord is constant and equal to k times of length of latus rectum, where k =    <

In any parabola the harmonic mean of intercepts of any focal chord is constant and equal to k times of length of latus rectum, where k =  

 

Option: 1

1


Option: 2

2


Option: 3

\mathrm{1/2}


Option: 4

4


Answers (1)

best_answer

Let the parabola be \mathrm{y^2=4 a x} and PQ be a focal chord if \mathrm{P \equiv\left(a t^2, 2 a t\right)}, then

\mathrm{Q \equiv\left(\frac{a}{t^2}, \frac{-2 a}{t}\right) .}         \mathrm{\text { Now } \begin{aligned} \text { PS } & =a+a t^2 \\ Q S & =a+\frac{a}{t^2} \end{aligned}}                 

\mathrm{\frac{2 P S . Q S}{P S+Q S}=\frac{2\left(a+a t^2\right) \cdot\left(a+\begin{array}{c} a \\ t^2 \end{array}\right)}{\left(a+a t^2\right)+\left(a+\frac{a}{t^2}\right)}}

\mathrm{Now =2 a}

which is independent of t and equal to length of semi latus rectum. 

 

 

 

Posted by

SANGALDEEP SINGH

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