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  Q1 If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, the locus, the midpoint of PN is a/an


Q2

 The product of perpendiculars from any point on the  hyperbola   having general equation  to is asymptotes is equal to?




2 If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, the locus, the midpoint of PN is a/an


6 years ago

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Answers : (1)

										

Dear Zahid


let equation of hyperbola is   x2 -y2 =a2


 and equation os aymptote is  :   y=x   (slope  m=1)


let N(x1,x1) on the asymptote ,and a point P(asecθ ,atanθ) on the hyperbola and(h,k) is the mid point of PN


 h=(x1+asecθ)/2   or  asecθ =2h-x1 ...............1


k = (x1+atanθ)/2  or  atanθ =2k-x1 ................2


 


since PN is perpendiculat to aymptote so slope of PN =-1


 (k-x1)/(h-x1)=-1


 x1 =(h+k)/2


so put this value in equation 1 and 2


 asecθ=(3h-k)/2


 atanθ =(3k-h)/2


 


now square and subtract


 a2(sec2θ -tan2θ) = {(3h-k)/2}2 - {(3k-h)/2}2


 a2 = {(3h-k)/2}2 - {(3k-h)/2}2


 4a2 =8(h2-k2)


 h2-k2=a2/2


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6 years ago

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