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  • The normal to the rectangular hyperbola     at the point  <img a

The normal to the rectangular hyperbola   \mathrm{x y=c^2}  at the point  \mathrm{{ }^{\prime} t_1{ }^{\prime}} meets the curve

again at the point \mathrm{' t_2^{\prime} \text { '. }}  Then the value of  \mathrm{t_1^3 t_2}  is

Option: 1

1


Option: 2

c


Option: 3

-c


Option: 4

-1


Answers (1)

best_answer

Normal at  \mathrm{t_1}  meets the curve again at \mathrm{t_2} .

So normal passes through point  \mathrm{\left(c t_1, c / t_1\right) \text { and }\left(c t_2, c / t_2\right)} .

Slope of normal = \mathrm{\frac{c / t_2-c / t_1}{c t_2-c t_1}}

                          =  \mathrm{-\frac{1}{t_1 t_2}}

Slope of normal from equation of normal at \mathrm{t_1=t_1^2}

So    \mathrm{t_1^2=-\frac{1}{t_1 t_2} \Rightarrow t_1^3 t_2=-1} .

 

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