Question

the locus of a point P which divides the line segment joining(1,0) and (2costheta,2 sintheta) internally in the ratio 1:2 for all theta is a

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ \mathrm{m}:\mathrm{n}=1:2 \\ \left(1,0\right)\mathrm{and}\left(2\mathrm{cos\theta },2\mathrm{sin\theta }\right) \\ \mathrm{so}\mathrm{using}\mathrm{internal}\mathrm{division}\mathrm{formula} \\ \mathrm{x}=\frac{2\mathrm{cos\theta }+2}{1+2},\mathrm{y}=\frac{2\mathrm{sin\theta }+0}{1+2} \\ \mathrm{cos\theta }=\frac{3\mathrm{x}-2}{2},\mathrm{and}\mathrm{sin\theta }=\frac{3\mathrm{y}}{2} \\ \mathrm{Squaring}\mathrm{and}\mathrm{adding} \\ {\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }={\left(\frac{3\mathrm{x}-2}{2}\right)}^2+\frac{9{\mathrm{y}}^2}{4} \\ \frac{{\left(3\mathrm{x}-2\right)}^2}{4}+\frac{9{\mathrm{y}}^2}{4}=1 \\ {\left(3\mathrm{x}-2\right)}^2+9{\mathrm{y}}^2=4 \\ \mathrm{Regards} \end{gathered}$$