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  • If the parabolas<img alt="\mathrm{ y^2=4 b(x-c) \ and \ y^2=8 a x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20y%5E2%3D4%20b%28x-c%29%20%5C%20and%20%5C%20y%5E2%3D8%20a%20x%

If the parabolas\mathrm{ y^2=4 b(x-c) \ and \ y^2=8 a x} have a common normal (other than x-axis), then which one of the following is a valid choice for the ordered triad (a, b, c) ?

Option: 1

(1,1,3)


Option: 2

(1,1,0)


Option: 3

\left(\frac{1}{2}, 2,3\right)


Option: 4

\left(\frac{1}{2}, 2,0\right)


Answers (1)

best_answer

Normal to the two given curves are 

\mathrm{\begin{aligned} & y=m(x-c)-2 b m-b m^3 \\ & y=m x-4 a m-2 a m^3 \end{aligned}}

If they have a common normal, then 

\mathrm{(c+2 b) m+b m^3=4 a m+2 a m^3}

\mathrm{\text { Now, }(4 a-c-2 b) m=(b-2 a) m^3}

We get that all the options are correct for m=0 i.e., when common normal is x-axis.
\mathrm{\text { For } m \neq 0}

\mathrm{m^2=\frac{4 a-c-2 b}{b-2 a}>0}

only option (a) satisfy it.

 

Posted by

seema garhwal

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