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  • If two different tangents of are the normals to <img alt="\mathrm{x^{2}=4by

If two different tangents of \mathrm{y^{2}=4 x} are the normals to \mathrm{x^{2}=4by}, then

Option: 1

\mathrm{|b|>\frac{1}{2 \sqrt{2}}}


Option: 2

\mathrm{|b|<\frac{1}{2 \sqrt{2}}}


Option: 3

\mathrm{|b|>\frac{1}{\sqrt{2}}}


Option: 4

\mathrm{|b|<\frac{1}{\sqrt{2}}}


Answers (1)

best_answer

Tangent to \mathrm{y^{2}=4 x} in terms of \mathrm{m} is

\mathrm{y=m x+\frac{1}{m}}.
Normal to \mathrm{x^{2}=4by} in terms of \mathrm{m} is
\mathrm{y=m x+2 b+\frac{b}{m^{2}}}.

If these are same lines, then
\mathrm{\frac{1}{m}=2 b+\frac{b}{m^{2}}}
\mathrm{\Rightarrow \quad 2 b m^{2}-m+b=0}

For two different tangents, we must have
\begin{gathered} 1-8 b^{2}>0 \\ \Rightarrow \quad|b|<\frac{1}{\sqrt{8}} \end{gathered}

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mansi

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