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  • If equation <img alt="\mathrm{y=m x+c\, and \, x \cos \alpha+y \sin \alpha=p}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7By%3Dm%20x&plus;c%5C%2C%20and%20%5C%2C%20x%20%5Cco

If equation \mathrm{y=m x+c\, and \, x \cos \alpha+y \sin \alpha=p} represent the same line, then :

Option: 1

\mathrm{c=p \sqrt{1+m^{2}}}


Option: 2

\mathrm{p=c \sqrt{1+m^{2}}}


Option: 3

\mathrm{p=c \sqrt{1-m^{2}}}


Option: 4

\mathrm{c=p \sqrt{1-m^{2}}}


Answers (1)

best_answer

Since the two equations represents the same line therefore

\mathrm{\frac{m}{\cos a}=\frac{-1}{\sin a}=\frac{c}{-p}}
\mathrm{\frac{m}{\cos a}=\frac{-c}{p} \Rightarrow c \cos \alpha=-m p}

\mathrm{ \cos a \frac{-m p}{c}\: and \: -\frac{1}{\sin a}=-\frac{c}{p}}

\mathrm{\Rightarrow c \cos \alpha=p \Rightarrow \sin \alpha=\frac{p}{c}}

\mathrm{\cos ^{2} a+\sin ^{2} a=1}
\mathrm{\therefore\left(\frac{m p}{c}\right)^{2}+\left(\frac{p}{c}\right)^{2}=1}
\mathrm{\Rightarrow m^{2} p^{2}+p^{2}=c^{2} \Rightarrow c^{2}=p^{2}\left(1+m^{2}\right)}
\mathrm{P c=p \sqrt{1+m^{2}}}

Hence (A) is the correct answer.

Posted by

himanshu.meshram

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