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Question Number 57174 by Tinkutara last updated on 31/Mar/19

Answered by ajfour last updated on 31/Mar/19

$$\mathrm{eq}.\mathrm{of}\mathrm{L}$$$$\mathrm{y}=3+\mathrm{rsin}\theta ,\mathrm{x}=2+\mathrm{rcos}\theta$$$$\mathrm{Q}\equiv (2+\mathrm{Rcos}\theta ,3+\mathrm{Rsin}\theta )\equiv (\mathrm{h},\mathrm{k})$$$$3+{\mathrm{r}}_{\mathrm{A}}\sin \theta =12+{\mathrm{r}}_{\mathrm{A}}\cos \theta$$$$\Rightarrow {\mathrm{r}}_{\mathrm{A}}=\frac{9}{\sin \theta -\cos \theta }$$$${\mathrm{r}}_{\mathrm{B}}=\frac{19}{\sin \theta -\cos \theta }$$$$\mathrm{As}{\mathrm{r}}_{\mathrm{A}}{\mathrm{r}}_{\mathrm{B}}={\mathrm{R}}^2$$$$\Rightarrow \frac{171}{{(\sin \theta -\cos \theta )}^2}={\mathrm{R}}^2$$$$\mathrm{or}{(\mathrm{k}-3-\mathrm{h}+2)}^2=171$$$$\Rightarrow \mathrm{locus}\mathrm{is}{(\mathrm{y}-\mathrm{x}-1)}^2=171.$$

Commented by Tinkutara last updated on 31/Mar/19

Thank you so much Sir!

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