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Q.

If xi,yi, i=1,2,3,4 are concyclic points on xy=4 then x1x2x3x4=

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a

–4 
 

b

16

c

–16 
 

d

4

answer is D.

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Detailed Solution

(i)

Given hyperbola xy=4

let P(xi, yi)=2ti, 2ti be any point on given hyperbola.

since given points are concyclic points then they lie on the circle x2+y2+2gx+2fy+k=0

4ti2+4ti2+2g(2ti)+2f26i+k=0 4ti4+4gti3+kti2+4fti+4=0 let four roots be t1, t2, t3, t4. Now t1 t2 t3 t4=44=1 t1 t2 t3 t4=1 Now x1 x2 x3 x4=(2t1) (2t2) (2t3) (2t4)                               =16(t1t2t3t4)                               =16.(1)                               =16.    

 

(ii)

Given hyperbola xy=c2

let P(xi, yi)=cti, cti where i=1, 2, 3, 4 be point on given hyperbola.

since given points are concyclic points then they lie on the circle x2+y2+2gx+2fy+k=0

c2ti2+c2ti2+2g(ti)+2fcti+k=0 c2ti4+2gcti3+kti2+2fcti+c2=0 It has four root, let it be t1, t2, t3, t4 Now t1t2t3t4=c2c2=1 Given y1 y2 y3 y4=4 Ct1 Ct2 Ct3 Ct4=4 C4t1 t2 t3 t4=4 C41=4 C4=4 c2=2.

 

(iii)

Given hyperbola xy=c2

x=c2y.

Given circle x2+y2=r2

sub 'x' value

c4y2+y2=r2 c4+y4=r2y2 y4-r2y2+c4=0 let y1, y2, y3, y4 be roots of this equation Now y1+y2+y3+y4=0.  sum of roots=-coefficient of y3coefficient of y4

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