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Q)

Asked by araima2001 14th February 2017, 10:07 AM
Answered by Expert
Answer:
begin mathsize 16px style straight P 1 space colon space straight y squared equals 4 straight a left parenthesis straight x minus straight lambda subscript 1 right parenthesis space and space straight P 2 space colon space straight x squared equals 4 straight a left parenthesis straight y minus straight lambda subscript 2 right parenthesis
Let space their space point space of space contact space be space straight A left parenthesis straight m comma straight n right parenthesis.
So comma space equation space of space the space tangent space to space straight P 2 space at space straight A colon
straight y subscript straight n equals 2 straight a left parenthesis straight x minus straight lambda subscript 1 plus straight m right parenthesis space.... left parenthesis 1 right parenthesis
and space equation space of space the space tangent space to space straight P 1 space at space straight A colon
straight x subscript straight m equals 2 straight a left parenthesis straight y minus straight lambda subscript 2 plus straight n right parenthesis space.... left parenthesis 2 right parenthesis
Simplifying space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get
2 ax minus straight y subscript straight n equals 2 straight a left parenthesis straight lambda subscript 1 minus straight m right parenthesis space..... left parenthesis 3 right parenthesis
and space 2 ay minus straight x subscript straight m equals 2 straight a left parenthesis straight lambda subscript 2 minus straight n right parenthesis space..... left parenthesis 4 right parenthesis
Since space left parenthesis 3 right parenthesis space and space left parenthesis 4 right parenthesis space are space identical space in space nature comma space by space comparing space the space coefficients space of space straight x space and space straight y.
fraction numerator 2 straight a over denominator straight m end fraction equals fraction numerator straight n over denominator 2 straight a end fraction
rightwards double arrow mn equals 4 straight a squared
rightwards double arrow open parentheses straight m over 2 close parentheses squared minus open parentheses straight m over 2 close parentheses squared plus open parentheses straight n over 2 close parentheses squared minus open parentheses straight n over 2 close parentheses squared plus mn over 2 plus mn over 2 equals 4 straight a squared
rightwards double arrow open parentheses straight m over 2 close parentheses squared plus mn over 2 plus open parentheses straight n over 2 close parentheses squared minus open parentheses straight m over 2 close parentheses squared minus open parentheses straight n over 2 close parentheses squared plus mn over 2 equals 4 straight a squared
rightwards double arrow open parentheses fraction numerator straight m plus straight n over denominator 2 end fraction close parentheses squared minus open parentheses fraction numerator straight m minus straight n over denominator 2 end fraction close parentheses squared equals 4 straight a squared
Replacing space fraction numerator straight m plus straight n over denominator 2 end fraction space by space straight X space and space fraction numerator straight m minus straight n over denominator 2 end fraction space by space straight Y space and space dividing space throughout space by space 4 straight a squared comma space we space get
rightwards double arrow fraction numerator straight X squared over denominator 4 straight a squared end fraction minus fraction numerator straight Y squared over denominator 4 straight a squared end fraction equals 1
which space represent space the space equation space of space straight a space Hyperbola
end style
Answered by Expert 16th February 2017, 3:54 PM
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