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If the four points (Mr, 1/Mr) where r=1,2,3,4 are concyclic then prove that product of their ordinate is 1.

If the four points (Mr, 1/Mr) where r=1,2,3,4 are concyclic then prove that product of their ordinate is 1.

Grade:11

1 Answers

Shukant Pal
25 Points
6 years ago
This question seems very hard but infact is very simple. We just have to assume the circle and put the coordinates (Mr, 1/Mr) into the equation and get a biquadratic equation.
 
Let’s assume the equation of the circle on which the given four points lie to be – x2 + y2 + 2gx + 2fy + c = 0.
 
Now putting the coordinates in this →
(Mr)2 + (1/Mr)2 + 2gMr + 2f/Mr + c = 0, and multiplying by Mr2,
(Mr)2 + 2gMr3 + cMr2 + 2fMr + 1 = 0. From this clearly → M1M2M3M4 = 1 or 1/(M1M2M3M4) = 1 or product of ordinates is 1.

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