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  • The line given by <img alt="\mathrm{(2 \cos \theta+3 \sin \theta) \mathrm{x}+(3 \cos \theta-5 \sin \theta) \mathrm{y}-(5 \cos \theta-2 \sin \theta)=0}" src="https://entrancecorner.oncodecogs.c

The line given by \mathrm{(2 \cos \theta+3 \sin \theta) \mathrm{x}+(3 \cos \theta-5 \sin \theta) \mathrm{y}-(5 \cos \theta-2 \sin \theta)=0} pass through a fixed point for all values of \mathrm{\Theta }. Find the coordinates of its reflection in the line \mathrm{x+y=\sqrt{2}.}

Option: 1

\mathrm{(1, \sqrt{2})}


Option: 2

\mathrm{(\sqrt{2}+1, \sqrt{2}+1)}


Option: 3

\mathrm{(\sqrt{2}-1, \sqrt{2}-1)}


Option: 4

none of these 


Answers (1)

best_answer

The given equation can be written as \mathrm{(2 x+3 y-5) \cos \theta+(3 x-5 y+2) \sin \theta=0} or \mathrm{(2 x+3 y-5)+\tan \theta(3 x-5 y+2)=0}
This passes through the point of intersection of the lines \mathrm{2 x+3 y-5=0 \text\ { and }\ 3 x-5 y+2=0} for all values of \mathrm{\Theta }. The coordinates of the point \mathrm{P} of intersection are \mathrm{\left ( 1,1 \right )}. 
Let \mathrm{Q\left ( h,k \right )} be the reflection of \mathrm{P(1, 1)} in the line \mathrm{x+y=\sqrt{2} \ \ \ \ \ ..........\left ( i \right )}
Then \mathrm{PQ} is perpendicular to \mathrm{\left ( i \right )} and the mid-point of \mathrm{PQ} lies on \mathrm{\left ( i \right )}.
\mathrm{\therefore \quad \frac{\mathrm{k}-1}{\mathrm{~h}-1}=1\ \ \ \ \quad \Rightarrow \mathrm{k}=\mathrm{h}}
\mathrm{\text { and } \frac{\mathrm{h}+1}{2}+\frac{\mathrm{k}+1}{2}=\sqrt{2} \quad\ \ \ \ \ \ \ \ \Rightarrow \mathrm{h}=\mathrm{k}=\sqrt{2}-1}

 

 

Posted by

manish painkra

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