Question
If length of the two segments of focal chord to the parabola y2 = 8ax are 2 and 4, then the value of a is-
- 1/3
- 2/3
- 4/3
- 4
Hint:
find out the harmonic mean of the lengths of the segments and equate it with 1/2 times length of latus rectum.
The correct answer is: 2/3
a=2/3
given y2=8ax
=> a= 2a
Focus : (2a,0)
Length of latus rectum = 4a = 4(2a) = 8a
We know that the harmonic mean of the lengths (PS,QS) where S is the focus is equal to the half of latus rectum
=> 2/(1/PS+1/QS)= ½(8a)
Or
2.PS.QS/(PS+QS) = 4a
Here, let PS= 4 and QS = 2
This gives us 2.4.2/6=4a
a=2/3
the harmonic mean of the lengths (PS,QS) where S is the focus is equal to the half of latus rectum
the parametric coordinates of the ends of focal chords are (at^2, 2at) and (a/t^2,-2a/t)
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