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zahid rashid dar Grade: 11
        If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, the locus, the midpoint of PN is a/an 
8 years ago

Answers : (1)

Badiuddin askIITians.ismu Expert
147 Points

Dear Zahid

you have not maintion where is point P and N ,

ia m solving this question asuming P is on the parabola and N is on the asymptotes

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


 x1 =(h+k)/2

so put this value in equation 1 and 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)


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Askiitians Experts

8 years ago
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