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Asked by araima2001 | 11th Nov, 2016, 10:12: PM

Expert Answer:

begin mathsize 16px style Consider space straight a space focal space chord space passing space through space the space focus space straight S left parenthesis straight a comma 0 right parenthesis space and space
meeting space the space parabola space at space straight P left parenthesis at squared comma 2 at right parenthesis space and space space straight Q left parenthesis at subscript 1 squared comma 2 at subscript 1 right parenthesis
Now comma space
slope space of space QS space equals space slope space of space PS space left parenthesis Since space straight P minus straight S minus straight Q space are space points space on space the space focal space chord right parenthesis
rightwards double arrow fraction numerator 2 at subscript 1 minus 0 over denominator at subscript 1 squared minus straight a end fraction equals fraction numerator 2 at minus 0 over denominator at squared minus straight a end fraction
On space solving space we space get comma space straight t subscript 1 equals fraction numerator negative 1 over denominator straight t end fraction
So comma space straight Q identical to space open parentheses straight a open parentheses fraction numerator negative 1 over denominator straight t end fraction close parentheses squared comma 2 straight a open parentheses fraction numerator negative 1 over denominator straight t end fraction close parentheses close parentheses rightwards double arrow straight Q identical to open parentheses straight a over straight t squared comma fraction numerator negative 2 straight a over denominator straight t end fraction close parentheses
Given space that space length space of space the space focal space chord space QP space equals space straight c
rightwards double arrow open parentheses 2 at space minus fraction numerator negative 2 straight a over denominator straight t end fraction close parentheses squared plus open parentheses at squared space minus straight a over straight t squared close parentheses squared equals straight c squared
rightwards double arrow 4 straight a open parentheses straight t space plus 1 over straight t close parentheses squared plus straight a squared open parentheses straight t squared space minus 1 over straight t squared close parentheses squared equals straight c squared
On space solving space further space we space get comma
straight a open parentheses straight t space plus 1 over straight t close parentheses squared equals straight c space space space.... left parenthesis straight i right parenthesis
We space are space given space that space the space focal space chord space is space at space straight a space distance space straight b space from space the space vertex.
So comma space equation space of space the space focal space chord space comes space out space to space be
left parenthesis 2 straight t right parenthesis straight x minus open parentheses straight t squared minus 1 close parentheses straight y minus 2 at equals 0
So comma space length space of space the space focal space chord space equals space fraction numerator negative 2 at over denominator straight t squared plus 1 end fraction
rightwards double arrow straight t plus 1 over straight t equals fraction numerator negative 2 straight a over denominator straight b end fraction space space space space..... left parenthesis ii right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
straight a open parentheses fraction numerator negative 2 straight a over denominator straight b end fraction close parentheses squared equals straight c
rightwards double arrow fraction numerator 4 straight a cubed over denominator straight b squared end fraction equals straight c rightwards double arrow straight b squared straight c equals 4 straight a cubed
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Answered by Rebecca Fernandes | 13th Nov, 2016, 11:25: PM

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