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Question Number 29134 by Tinkutara last updated on 04/Feb/18

Commented by ajfour last updated on 04/Feb/18

Commented by ajfour last updated on 04/Feb/18

$$letPQ=r_1,PS=r_2,PR=r_3$$$$\frac{1}{r_2}=\frac{r_1+r_3}{2r_1{r}_3}...\left(i\right)$$$$eq.ofvariablelinethroughP:$$$$y=\beta +r\sin \theta ,x=\alpha +r\cos \theta$$$$Q({at}_1^2,2{at}_1),R({at}_2^2,2{at}_2)$$$$QandRlieonlineandparabola$$$$\Rightarrow 2at=\beta +r\sin \theta$$$${at}^2=\alpha +r\cos \theta$$$${(\beta +r\sin \theta )}^2=4a(\alpha +r\cos \theta )$$$$\Rightarrow r^2\sin {}^2\theta +(2\beta \sin \theta -4a\cos \theta )r+{\beta }^2-4a\alpha =0$$$$r_1,r_{3}arerootsofthiseq.$$$$\Rightarrow \frac{2r_1{r}_3}{r_1+r_3}=\frac{2({\beta }^2-4a\alpha )}{4a\cos \theta -2\beta \sin \theta }$$$$usingthisin\left(i\right)$$$$\frac{1}{r_2}=\frac{4a\cos \theta -2\beta \sin \theta }{2({\beta }^2-4a\alpha )}...\left(ii\right)$$$$or{\beta }^2-4a\alpha =2{ar}_2\cos \theta -\beta r_2\sin \theta$$$$butS(h,k)liesonline,so$$$$h=\alpha +r_2\cos \theta ,k=\beta +r_2\sin \theta$$$$usingthisin\left(ii\right)$$$${\mathit{\beta }}^2-4\mathit{a}\mathit{\alpha }=2\mathit{a}(\mathit{h}-\mathit{\alpha })-\mathit{\beta }(\mathit{k}-\mathit{\beta })$$$$\Rightarrow 2a(h+\alpha )=k\beta$$$$soreplacing(h,k)by(x,y)$$$$\mathit{y}=\left(\frac{2\mathit{a}}{\mathit{\beta }}\right)x+\frac{2\mathit{a}\mathit{\alpha }}{\mathit{\beta }}.$$$$\mathit{s}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{p}\mathit{e}\mathit{n}\mathit{d}\mathit{e}\mathit{n}\mathit{t}\mathit{o}\mathit{f}\mathit{\alpha }.$$

Commented by Tinkutara last updated on 05/Feb/18

Wonderful Sir! ���� Thanks! ��

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