The-base-of-triangle-passes-through-a-fixed-point-p-a-b-and-its-sides-are-respectively-bisected-at-right-angles-by-the-line-x-y-0-and-x-9y-if-the-locus-of-the-third-vartex-is-a-circle-then-find-its

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Question Number 43005 by mondodotto@gmail.com last updated on 06/Sep/18

$$\mathrm{The}\mathrm{base}\mathrm{of}\mathrm{triangle}\mathrm{passes}\mathrm{through}$$$$\mathrm{a}\mathrm{fixed}\mathrm{point}\mathrm{p}(\mathrm{a},\mathrm{b})\mathrm{and}\mathrm{its}\mathrm{sides}$$$$\mathrm{are}\mathrm{respectively}\mathrm{bisected}\mathrm{at}\mathrm{right}\mathrm{angles}$$$$\mathrm{by}\mathrm{the}\mathrm{line}\mathrm{x}+\mathrm{y}=0\mathrm{and}\mathrm{x}=9\mathrm{y}$$$$\mathrm{if}\mathrm{the}\mathrm{locus}\mathrm{of}\mathrm{the}\mathrm{third}\mathrm{vartex}\mathrm{is}\mathrm{a}\mathrm{circle}.$$$$\mathrm{then}\mathrm{find}\mathrm{its}\mathrm{equation}.$$

Commented by mondodotto@gmail.com last updated on 06/Sep/18

$$\mathrm{please}\mathrm{help}\mathrm{me}\mathrm{this}\mathrm{problem}$$

Commented by ajfour last updated on 06/Sep/18

Commented by ajfour last updated on 06/Sep/18

$$letA(h,k)$$$$\Rightarrow B(-k,-h)$$$$letC(\alpha ,\beta )$$$$reflectionofCiny=mx+cis$$$$\alpha =h-\frac{2m(c+mh-k)}{1+m^2}$$$$\beta =k+\frac{2(c+mh-k)}{1+m^2}$$$$herewehave\mathit{y}=\frac{\mathit{x}}{9}$$$$\Rightarrow m=\frac{1}{9},c=0$$$$\Rightarrow \alpha =h-\frac{\frac{2}{9}(\frac{h}{9}-k)}{1+\frac{1}{81}}$$$$\Rightarrow \mathit{\alpha }=\mathit{h}-\frac{9}{41}(\frac{\mathit{h}}{9}-\mathit{k})$$$$and\mathit{\beta }=\mathit{k}+\frac{81}{41}(\frac{\mathit{h}}{9}-\mathit{k})$$$$eq.oflinethroughBCis$$$$y+h=\frac{\beta +h}{\alpha +k}(x+k)$$$$AsthispassesthroughP(a,b)$$$$b+h=\frac{\beta +h}{\alpha +k}(a+k)$$$$replacing\alpha ,\beta intermsofh,k$$$$andchangingh,ktox,yweget$$$$thedesiredlocusofA\left(thirdvertex\right)$$$$\mathit{b}+\mathit{h}=\frac{\mathit{k}+\frac{81}{41}(\frac{\mathit{h}}{9}-\mathit{k})+\mathit{h}}{\mathit{h}-\frac{9}{41}(\frac{\mathit{h}}{9}-\mathit{k})+\mathit{k}}(\mathit{a}+\mathit{k})$$$$\Rightarrow (b+x)[x+y-\frac{9}{41}(\frac{x}{9}-y)]$$$$=(a+y)[x+y+\frac{81}{41}(\frac{x}{9}-y)]$$$$locusfoundis$$$$40({\mathit{x}}^2+{\mathit{y}}^2)-41(\mathit{b}-\mathit{a})(\mathit{x}+\mathit{y})$$$$-9(\mathit{a}\mathit{x}-\mathit{b}\mathit{y})=0.$$

Commented by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

$$dearajfoursirwhichapphelpstodraw\dots for$$$$examplethisproblem\dots$$

Commented by ajfour last updated on 06/Sep/18

$$lekhdiagram.$$$$ihaveunderstoodhowtouseit$$$$anyconfusion,ucanaskme..$$

Commented by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

$$okthankyou\dots ihavelatentwishtoseeourselves$$$$wearevirtuallyconnectedbutneverseeeachother$$$$soifadminofthisappgivepermissiontoupload$$$$ourpictureatleastonece\dots i$$

Commented by MJS last updated on 06/Sep/18

$$\mathrm{we}\mathrm{could}\mathrm{upload}\mathrm{a}\mathrm{self}\mathrm{portrait}\mathrm{including}\mathrm{a}$$$$\mathrm{math}\mathrm{example}\dots$$

Commented by mondodotto@gmail.com last updated on 06/Sep/18

$$\mathrm{thanx}$$

Commented by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

Commented by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

Commented by behi83417@gmail.com last updated on 06/Sep/18

$$wonderfulidea!$$$$waitingforsir:MrW3,ajfour,$$$$sirMJS,prop.Abdo,\dots ..$$

Commented by MJS last updated on 06/Sep/18

Commented by MJS last updated on 06/Sep/18

$$\mathrm{the}\mathrm{man}\mathrm{is}1.95\mathrm{m}\mathrm{tall},\mathrm{his}\mathrm{beard}\mathrm{is}\mathrm{half}\mathrm{as}\mathrm{long}$$$$\mathrm{as}\mathrm{Santa}{\mathrm{Claus}}^{\prime }.\mathrm{how}\mathrm{many}\mathrm{thorns}\mathrm{has}\mathrm{his}$$$$\mathrm{cactus}\mathrm{if}\mathrm{the}\mathrm{temperature}\mathrm{is}35°\mathrm{C}?$$

Commented by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

$$excellent\dots sir\dots ihavethoughtavirtualimage$$$$ofyousir\dots ishallpost\dots whatiimaginedabout$$$$yousir\dots$$$$$$

Commented by MJS last updated on 06/Sep/18

$$\mathrm{ha}!{\mathrm{I}}^{\prime }\mathrm{d}\mathrm{like}\mathrm{to}\mathrm{see}\mathrm{it}!$$

Commented by math khazana by abdo last updated on 07/Sep/18

$$wearewaitingtoknowsirTinkutara,sirAjfour,sirMrw,sir$$$$Raul,sirPrakachjain,sirSmai3\dots$$

Commented by malwaan last updated on 07/Sep/18

Commented by malwaan last updated on 07/Sep/18

$$\mathrm{that}\mathrm{is}\mathrm{me}$$

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Sep/18

$$thankyousir\dots$$

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