Hyperbola And Parabola
Geometry
Level
4
A point moves such that sum of slopes of the normals drawn from it to the hyperbola xy=16 is equal to the sum of ordinates of feet of normals. The locus of P is a curve C.
The equation of the curve C is:
x^2=16y
Y^2=16x
Y^2=8x
x^2=12y
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
first we write the equation of a normal on any point * ( a , b ) * on hyperbola
ie x a − y b = a ( 2 ) − b ( 2 )
now let the point from which we draw normal be (h,k)
h a − y b = a ( 2 ) − b ( 2 ) slope=a/b=16/{b^(2)} now since question is concerned about ordinate we eliminate a by using
a ∗ b = 1 6 now the equation we obtain is biquadratic in b. so we find sum of roots ie sum of ordinate.
to find sum of slope we form the equation with roots 1/b^(2) using transformation of equation and find sixteen times sum of roots. and then just equate we get h ( 2 ) = 1 6 k so locus is x ( 2 ) = 1 6 y