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Maths-

General

Easy

Question

If $ell$ ₁ & $ell$ ₂ are length of segments of focal chord of parabola y² = 4ax than harmonic mean of $ell$ ₁ & $ell$ ₂ is equal to

4a
3a
2a
a

The correct answer is: 2a

To find the harmonic mean of the segments of a focal chord.
Coordinates of focal chord
A= $open parentheses a t subscript 1 squared comma 2 a t subscript 1 close parentheses$
B= $open parentheses a t subscript 2 squared comma 2 a t subscript 2 close parentheses$
Harmonic mean = $fraction numerator 2 over denominator begin display style 1 over A end style plus begin display style 1 over B end style end fraction equals fraction numerator 2 over denominator begin display style fraction numerator 1 over denominator a t subscript 1 squared plus a end fraction end style plus begin display style fraction numerator 1 over denominator a t subscript 2 squared plus a end fraction end style end fraction space equals space fraction numerator 2 a over denominator fraction numerator 1 over denominator t subscript 1 squared plus 1 end fraction plus fraction numerator 1 over denominator t subscript 2 squared plus 1 end fraction end fraction$
$t subscript 2 equals negative 1 over t subscript 1 rightwards double arrow fraction numerator 2 a over denominator begin display style fraction numerator 1 plus t squared over denominator t squared plus 1 end fraction end style end fraction space equals space 2 a$

Therefore, harmonic mean of the segments of a focal chord is 2a

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