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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
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7 years ago

147 Points
```										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
Please feel free to post as many doubts on our discussion forum as you can.If you find any question Difficult to understand - post it here and we will get you the answer and detailed  solution very  quickly. We are all IITians and here to help you in your IIT JEE preparation.All the best. Regards,Askiitians ExpertsBadiuddin

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7 years ago
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