Q. 31 - Calculus

Question

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31. 

$r=\frac{\alpha }{1+\sin \theta }$

Step-by-Step Solution

Answer

The equation of the parabola is $x^2=-2y\alpha +{\alpha }^2$.

Step 1. Given information.

The given polar equation is $r=\frac{\alpha }{1+\sin \theta }$.

Step 2. Conversion.

Converting the polar into cartesian co-ordinates,

$$\begin{gathered} r=\frac{\alpha }{1+\sin \theta } \\ r(1+\sin \theta )=\alpha \\ r+r\sin \theta =\alpha \\ \sqrt{x^2+y^2}+r·\frac{y}{r}=\alpha \left[\text{ since }r=\sqrt{x^2+y^2}\text{ and }r\sin \theta =y\right] \\ \sqrt{x^2+y^2}+y=\alpha \end{gathered}$$

Step 3. Final answer.

On simplifying the equation,

$$\begin{gathered} \sqrt{x^2+y^2}+\backslash \mathrm{not}y-\backslash \mathrm{not}y=\alpha -y \\ \sqrt{x^2+y^2}=\alpha -y \\ {\left(\sqrt{x^2+y^2}\right)}^2=(\alpha -y{)}^2 \\ {x}^2+y^2={\alpha }^2+y^2-2y\alpha \\ {x}^2+y^2-y^2={\alpha }^2+y^2-2y\alpha -y^2 \\ {x}^2=-2y\alpha +{\alpha }^2 \end{gathered}$$