$r\=\frac{5\/2}{1\+\left(1\/2\right) \cos \left(\mathrm{\θ}\right)}$ 

the values $e\=1\/2$ and $d\=5$ can be immediately deduced. Hence, it follows that

$\mathrm{plots}:-\mathrm{implicitplot}\left(r\=5\/\left(2\+\cos \left(\mathrm{\θ}\right)\right)\,r\=0..5\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coords}\=\mathrm{polar}\,\mathrm{scaling}\=\mathrm{constrained}\,\mathrm{gridrefine}\=3\right)$

$r\=\frac{5}{2\+\cos \left(\mathrm{\theta }\right)}$

$r\=\frac{5}{2\+\cos \left(\mathrm{\θ}\right)}$

$\stackrel{\text{evaluate at point}}{\to }$

$\sqrt{x^2\+y^2}\=\frac{5}{2\+\frac{x}{\sqrt{x^2\+y^2}}}$

$\stackrel{\text{cross multiply}}{\to }$

$\sqrt{x^2\+y^2}\left(2\sqrt{x^2\+y^2}\+x\right)\=5\sqrt{x^2\+y^2}$

Figure 7.1.12(c)   The Equation Manipulator

Step

Result

Multiply equation by $1\/\sqrt{x^2\+y^2}$

$2\sqrt{x^2\+y^2}\+x\=5$

Add $-x$ to both sides

$2\sqrt{x^2\+y^2}\=5-x$

Square both sides

$4x^2\+4y^2\={\left(5-x\right)}^2$

Group terms on left side

$4x^2\+4y^2-{\left(5-x\right)}^2\=0$

Apply simplify to left side

$3x^2\+4y^2\+10x-25\=0$

Complete the square on the left side

$3{\left(x\+\frac{5}{3}\right)}^2\+4y^2-\frac{100}{3}\=0$

Add $100\/3$ to equation

$3{\left(x\+\frac{5}{3}\right)}^2\+4y^2\=\frac{100}{3}$

Multiply equation by $3\/100$

$\frac{9}{100}{\left(x\+\frac{5}{3}\right)}^2\+\frac{3}{25}{y}^2\=1$

Table 7.1.12(a)   Application of the Equation Manipulator