Question

​the fix point through which the line (2cosθ+3sinθ)x + (3cosθ-5sinθ)y - (5cosθ-2sinθ)=0 passes for all values of θ is

Answer 1

Dear Student,

$$\begin{gathered} (2\cos \theta +3\sin \theta )x+(3\cos \theta -5\sin \theta )y-(5\cos \theta -2\sin \theta )=0 \\ \Rightarrow (2\cos \theta +3\sin \theta )x+(3\cos \theta -5\sin \theta )y=(5\cos \theta -2\sin \theta ) \\ Throughtrialanderror,substitutex=1andy=1inaboveequation,weget \\ (2\cos \theta +3\sin \theta )1+(3\cos \theta -5\sin \theta )1=\cos \theta (2+3)+\sin \theta (3-5)=5\cos \theta -2\sin \theta \\ So,thepointx=1andy=1alwayssatisfiestheaboveequation.So,itwwilldefinetlypassthroughtthegivenline. \end{gathered}$$
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