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Question Number 207594 by MATHEMATICSAM last updated on 19/May/24

$$\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1$$$$x\sin \theta -y\cos \theta =\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta }$$$$\mathrm{Eliminate}\theta .$$

Answered by Frix last updated on 20/May/24

$$t=\tan \theta \mathrm{and}\mathrm{transforming}\mathrm{gives}$$$$A.\frac{x}{a}+\frac{ty}{b}=\sqrt{t^2+1}[\Rightarrow \frac{x}{a}+\frac{ty}{b}⩾0]$$$$B.tx-y=\sqrt{a^2{t}^2+b^2}[\Rightarrow tx⩾y]$$$$\mathrm{Squaring}[\mathrm{might}\mathrm{introduce}\mathrm{false}\mathrm{solutions}!]$$$$\mathrm{and}\mathrm{transforming}\mathrm{gives}$$$$t^2+\frac{2bxy}{a(y^2-b^2)}t+\frac{b^2(x^2-a^2)}{a^2(y^2-b^2)}=0[\Rightarrow y\ne \pm b]$$$$t^2-\frac{2xy}{x^2-a^2}t+\frac{y^2-b^2}{x^2-a^2}=0[\Rightarrow x\ne \pm a]$$$$\mathrm{Subtracting}\mathrm{and}\mathrm{solving}\mathrm{gives}$$$$t=-\frac{{bx}^2-{ay}^2-ab(a-b)}{2axy}$$$$\mathrm{Now}\mathrm{insert}\mathrm{this}\mathrm{in}A\mathrm{or}B$$$$$$$${\mathrm{I}}^{\prime }\mathrm{m}\mathrm{not}\mathrm{sure}\mathrm{if}\mathrm{this}\mathrm{is}\mathrm{a}\mathrm{valid}\mathrm{solution}.\mathrm{The}$$$$\mathrm{given}\mathrm{equations}\mathrm{define}\mathrm{two}\mathrm{lines}\mathrm{in}{\mathbb{R}}^2\mathrm{with}$$$$a,b,\theta \mathrm{as}\mathrm{constants}\mathrm{or}\mathrm{rather}\mathrm{strange}\mathrm{objects}$$$$\mathrm{in}{\mathbb{R}}^3\mathrm{with}a,b\mathrm{as}\mathrm{constants}...$$

Answered by mr W last updated on 20/May/24

$$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$$$$\cos \alpha \cos \theta +\sin \alpha \sin \theta =\frac{1}{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}}$$$$\cos (\theta -\alpha )=\frac{1}{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}}$$$$\theta =\alpha +{\cos }^{-1}\frac{1}{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}}$$$$\theta ={\cos }^{-1}\frac{x}{a\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}}+{\cos }^{-1}\frac{1}{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}}$$$$\Rightarrow \sin \theta =\frac{y}{b(\frac{x^2}{a^2}+\frac{y^2}{b^2})}+\frac{x\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}}{a(\frac{x^2}{a^2}+\frac{y^2}{b^2})}$$$$\Rightarrow \cos \theta =\frac{x}{a(\frac{x^2}{a^2}+\frac{y^2}{b^2})}-\frac{y\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}}{b(\frac{x^2}{a^2}+\frac{y^2}{b^2})}$$$$insertthisintoequation\left(ii\right)$$

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