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CBSE Class 11-science Answered

10.  A normal is drawn to the hyperbola  x^2 / a^2 - y^2 / b^2 = 1  at P which meets the transverse axis  at  G . If perpendicular from  G  on the asymptote meets  it  at  L , show that LP  is parallel to the conjugate axis.

Asked by sudhanshubhushanroy | 31 Mar, 2018, 11:02: AM
Expert Answer
begin mathsize 16px style Let space straight M open parentheses asecθ comma space btanθ close parentheses space be space any space point space on space hyperbola.
Equation space of space the space normal space at space straight P space is space axcosθ plus bycotθ equals straight a squared plus straight b squared
According space to space the space questio space normal space meets space the space straight x minus axis space left parenthesis transverse space axis right parenthesis
at space straight y equals 0
axcosθ plus bycotθ equals straight a squared plus straight b squared
Put space straight y equals 0
axcosθ equals straight a squared plus straight b squared
straight x equals fraction numerator straight a squared plus straight b squared over denominator straight a end fraction secθ
straight G equals open parentheses fraction numerator straight a squared plus straight b squared over denominator straight a end fraction secθ comma space 0 close parentheses
The space equation space of space line space perpendicular space to space the space asymptote space bx minus ay equals 0 space and space
passing space through space straight G.
Equation space of space GL space is space straight y equals fraction numerator negative straight a over denominator straight b end fraction open parentheses straight x minus fraction numerator straight a squared plus straight b squared over denominator straight a end fraction secθ close parentheses
ax plus by equals open parentheses straight a squared plus straight b squared close parentheses secθ
intersection space with space the space asymptote space bx minus ay equals 0 space gives space straight x equals asecθ
Hence comma space straight x space coordinate space of space straight L space is space asecθ comma space which space is space equal space to space the space straight x minus coordinate space of space the space point space straight M.
LP space is space parallel space to space straight y minus axis. space Hence comma space LP space is space parallel space to space conjugate space axis. end style
Answered by Sneha shidid | 02 Apr, 2018, 10:09: AM
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