Further Graphing Using Polar Coordinates - Questions

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Question 1

58630

The curve with a polar equation $r=1-\cos \theta$ is a Cardioid.

i) Sketch the graph of the Cardioid

ii) Find the Cartesian equation

Answer

i) Refer to worked solutions ii) $(x^2+y^2+x)^2=x^2+y^2$

Worked Solution

$$
\begin{aligned}\text { (ii) }\quad
x &=r \cos \theta \\
\frac{x}{r} &=\cos \theta \\
r &=1-\frac{x}{r} \\
r^{2} &=r-x\\
x^{2}+y^{2}&=\sqrt{x^{2}+y^{2}}-x\\
x^{2}+y^{2}+x&=\sqrt{x^{2}+y^{2}}\\
\left(x^{2}+y^{2}+x\right)^{2}&=x^{2}+y^{2}
\end{aligned}
$$

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