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  • If the line lx + my + n = 0 is normal to the hyperbola xy = 1, then Option: 1 l>0, m>0Option

If the line lx + my + n = 0 is normal to the hyperbola xy = 1, then 

Option: 1

l>0, m>0


Option: 2

l>0, m<0


Option: 3

l<0, m<0


Option: 4

both (a) and (c)


Answers (1)

best_answer

 

 

Properties of rectangular Hyperbola -

Properties of rectangular Hyperbola:

 

\\\mathrm{(i)\;\;\;\text{The parametric equation of the rectangular hyperbola}\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;are\;\;x=ct\;\;and\;\;y=\frac{c}{t}.}\\\\\mathrm{(ii)\;\;\;\text{The equation of the tangent to the rectangular hyperbola}\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;at\;(x_1,y_1)\;\;is\;\;x y_{1}+x_{1} y=c^{2}.}\\\\\mathrm{(iii)\;\;\;\text { The equation of the tangent at }\left(c t, \frac{c}{t}\right) \text { to the hyperbola }\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;\text { is } \frac{x}{t}+y t=2 c.}\\\\\mathrm{(iv)\;\;\;\text { The equation of the normal at }\left(x_{1}, y_{1}\right) \text { to the hyperbola }\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;\text { is } x x_{1}-y y_{1}=x_{1}^{2}-y_{1}^{2}.}\\\\\mathrm{(v)\;\;\;\text { The equation of the normal at }t \text { to the hyperbola }\;xy=c^2}\\\mathrm{\;\;\;\;\;\;\;\;\;\text { is } x t^{3}-y t-c t^{4}+c=0.}

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Equation of normal at (t, 1/t) is 

\\xt^3-yt-t^4+1=0\\\text{slope is }t^2=-\frac{l}{m}\\-\frac{l}{m}>0\Rightarrow \frac{l}{m}<0\\\therefore l>0,m<0\\or\;\;l<0,m>0

Posted by

himanshu.meshram

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