Conic-Sections Question 581
Question: If the coordinates of four concyclie points on the rectangular hyperbola $ xy=c^{2} $ are $ (ct_{i},c/t_{i}),i=1,2,3,4 $ then
Options:
A) $ t_1t_2t_3t_4=-1 $
B) $ t_1t_2t_3t_4=1 $
C) $ t_1t_3=t_2t_4 $
D) $ t_1+t_2+t_3+t_4=c^{2} $
Show Answer
Answer:
Correct Answer: B
Solution:
[b] Let the points lie on the circle $ x^{2}+y^{2}+2gx+2fy+2fy+k=0, $ Then $ c^{2}t_i^{2}+\frac{c^{2}}{t_i^{2}}+2get_{i}+2f\frac{c}{t_{i}}+k=0 $
$ \Rightarrow c^{2}t_i^{4}+2gct_i^{3}+kt_i^{3}+2fct_{i}+c^{2}=0 $ Its roots are $ t_1,t_2,t_3,t_4 $ so $ t_1t_2t_3t_4=\frac{c^{2}}{c^{2}}=1 $ Also, $ t_1+t_2+t_3+t_4=-\frac{2gc}{c^{2}}=-\frac{2g}{c} $
BETA
