If ( a , (1/ a )),( b , (1/ b )),( c , (1/ c )) &( d , (1/ d )) are four distinct points on a circle of radius 4 units then, abcd =

Question

If ( a , (1/ a )),( b , (1/ b )),( c , (1/ c )) &( d , (1/ d )) are four distinct points on a circle of radius 4 units then, abcd = (A) 4 (B) 1 / 4 (C) 1 (D) 16

Tardigrade Admin · Accepted Answer

Let equation of the circle be x2+y2+2 g x+2 f y+λ=0 (t, (1/t)) be a point on the circle ∴ t 2+(1/ t 2)+2 gt +2 f (1/ t )+λ=0 t 4+2 gt 3+λ t 2+2 ft +1=0 roots of the above equation are a , b , c , d ∴ abcd =1

Q. If $(a, \frac{1}{a}), (b, \frac{1}{b}), (c, \frac{1}{c}) & (d, \frac{1}{d})$ are four distinct points on a circle of radius 4 units then, abcd $=$

4

$1/4$

1

16

Let equation of the circle be
$x^{2} + y^{2} + 2 gx + 2 f y + λ = 0$
$(t, \frac{1}{t})$ be a point on the circle
$∴ t^{2} + \frac{1}{t ^{2}} + 2 g t + 2 f \frac{1}{t} + λ = 0$
$t^{4} + 2 g t^{3} + λ t^{2} + 2 f t + 1 = 0$
roots of the above equation are $a, b, c, & d$
$∴ ab c d = 1$

Q. If (a,a1​),(b,b1​),(c,c1​)&(d,d1​) are four distinct points on a circle of radius 4 units then, abcd =

4

1/4

1

16

Let equation of the circle be x2+y2+2gx+2fy+λ=0 (t,t1​) be a point on the circle ∴t2+t21​+2gt+2ft1​+λ=0 t4+2gt3+λt2+2ft+1=0 roots of the above equation are a,b,c,&d ∴abcd=1

Q. If $(a, \frac{1}{a}), (b, \frac{1}{b}), (c, \frac{1}{c}) & (d, \frac{1}{d})$ are four distinct points on a circle of radius 4 units then, abcd $=$

$1/4$

Let equation of the circle be
$x^{2} + y^{2} + 2 gx + 2 f y + λ = 0$
$(t, \frac{1}{t})$ be a point on the circle
$∴ t^{2} + \frac{1}{t ^{2}} + 2 g t + 2 f \frac{1}{t} + λ = 0$
$t^{4} + 2 g t^{3} + λ t^{2} + 2 f t + 1 = 0$
roots of the above equation are $a, b, c, & d$
$∴ ab c d = 1$