A point P moves such that sums of the slopes of the normal drawn from it to the hyperbola xy=16 - Maths - Conic Sections

Question

A point P moves such that sums of the slopes of the normal drawn
from it to the hyperbola xy=16 is equal to the sum of ordinates of
feet of normal The locus of point P is a curve C Find the equation
of curve C

Ajanta Trivedi · Accepted Answer

let P(h,k) be a point in the plane of
the hyperbola xy = 16 = 4^2
the equation of the normal at the point (4t , 4/t ) to the
hyperbola xy = 4^2 is
xt3-yt-4t4+4=0if it passes through P(h,k) , thenht3-kt-4t4+4=0⇒4t4-ht3+kt-4=0 
(1)
this is a fourth degree equation in t so it gives four values of t
sat t1 , t2 , t3 ,
t4
corresponding to each value of t there is a point on the hyperbola
such that the normal at it passes through P(h,k)
let the four points be
Act1 ,ct1 , Bct2 ,ct2 , Cct3 ,ct3 and Dct4 ,ct4
such that normal at these points pass through P(h,k)
since t1 , t2 ,t3 ,t4 are roots of equation (1)
therefore,t1+t2+t3+t4=hc=h4  (2)Σ t1t2=0 
(3)Σt1t2t3=-k4  (4)t1t2t3t4=-1  (5)
it is given that the sum of the slopes of the normals at A, B , C
and D is equal to the sum of the ordinates of these points
therefore
t12+t22+t32+t42=ct1+ct2+ct3+ct4=4t1+4t2+4t3+4t4(t1+t2+t3+t4)2-2 
Σ t1t2=4
Σt1t2t3t1t2t3t4⇒h242-2*0=4*-k/4-1h242=k⇒h2=16*kso the locus of (h,k) is x2=16y , which is a parabola
hope this helps you

A point P moves such that sums of the slopes of the normal drawn from it to the hyperbola xy=16 is equal to the sum of ordinates of feet of normal. The locus of point P is a curve C. Find the equation of curve C.