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Question Number 211235 by behi834171 last updated on 01/Sep/24

Commented by behi834171 last updated on 01/Sep/24

$$\mathrm{\angle }\mathit{A}\mathit{B}\mathit{C};isgiven.$$$$\mathit{D};isapointasitshowen,and$$$$\mathit{D}\mathit{G}\mathrm{\&}\mathit{D}\mathit{F};aredistanceof:\mathit{D},from:$$$$\mathit{A}\mathit{B}\mathrm{\&}\mathit{B}\mathit{C},suchthat:\frac{\mathit{D}\mathit{G}}{\mathit{D}\mathit{F}}=\mathit{k}\left(\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{t}\right)$$$$\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{t}\mathit{h}\mathit{e}\mathit{l}\mathit{o}\mathit{c}\mathit{u}\mathit{s}\mathit{o}\mathit{f}:\mathit{D}.$$

Answered by A5T last updated on 01/Sep/24

Commented by A5T last updated on 01/Sep/24

$$Let\mathrm{\angle }ABC=\mathrm{\angle }GBF=\theta$$$$\mathrm{\angle }GBD=\beta \Rightarrow \mathrm{\angle }DBF=\theta -\beta$$$$\frac{sin90°}{BD}=\frac{sin\beta }{DG}=\frac{sin(\theta -\beta )}{DF}\Rightarrow \frac{sin\beta }{sin(\theta -\beta )}=k$$$$Since\theta andkareknown,wecansolvefor\beta$$$$\Rightarrow Thelocusofpointsisalinewhichmakes$$$$anangleof\beta withlineAB.$$

Commented by behi834171 last updated on 01/Sep/24

$$ThankyousirAST.$$$$yes.thelocusisalinecrossesatBpoint.$$

Answered by mr W last updated on 01/Sep/24

Commented by mr W last updated on 01/Sep/24

$$\alpha +\beta =\theta$$$$\sin \alpha =\frac{DG}{BD}$$$$\sin \beta =\frac{DF}{BD}$$$$\frac{\sin \alpha }{\sin \beta }=\frac{DG}{DF}=k$$$$\Rightarrow \sin \alpha =k\sin \beta =k\sin (\theta -\alpha )$$$$\Rightarrow \sin \alpha =k(\sin \theta \cos \alpha -\cos \theta \sin \alpha )$$$$\Rightarrow \frac{\sin \theta }{\tan \alpha }-\cos \theta =\frac{1}{k}$$$$\Rightarrow \tan \alpha =\frac{\sin \theta }{\cos \theta +\frac{1}{k}}$$$$\Rightarrow \alpha ={\tan }^{-1}\left(\frac{\sin \theta }{\cos \theta +\frac{1}{k}}\right)=constant$$$$\Rightarrow locusofpointDisalineBDwith$$$$anangle\alpha tothelineAB.$$

Commented by behi834171 last updated on 01/Sep/24

$$Thankyousomuchdearmaster.$$$$$$If we have just two intersecting lines,inested of given angle:ABC; is the locus unique?

Commented by mr W last updated on 01/Sep/24

$$inthiscasethepointcanbeevery$$$$whereexcept(0,0).$$

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