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Question

If the normals at (xi,yi), where, i=1,2,3,4 on the rectangular hyperbola xy=c2 meet at (α,β). and x21+x22+x23+x24 is a and y21+y22+y23+y24 is b, then a+b is
  1. α+β
  2. 2(α+β)
  3. α2+β2
  4. 2(α2+β2)


Solution

The correct option is C α2+β2
The equation of the normal to the hyperbola xy=c2 at (ct,ct) is
xt3ytct4+c=0
which is passing through (α,β)
Thus, ct4αt3+βtc=0.
Let its four roots are t1,t2,t3,t4.

t1=αc,t1t2=0(t1t2t3)=βc
and (t1t2t3t4)=1.
x21+x22+x23+x24=c2[(t1)22t1t2]a=α2

c(1t1+1t2+1t3+1t4)=y1+y2+y3+y4=βy21+y22+y23+y24=β22c2(1/t1t2)b=β2

a+b=α2+β2

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