Answer:
[tex]y = - \frac{ 1 }{10} {x}^{2} + \frac{5}{2} [/tex]
Step-by-step explanation:
We want to prove algebraically that:
[tex]r = \frac{10}{2 + 2 \sin \theta} [/tex]
is a parabola.
We use the relations
[tex] {r}^{2} = {x}^{2} + {y}^{2} [/tex]
and
[tex]y = r \sin \theta[/tex]
Before we substitute, let us rewrite the equation to get:
[tex]r(2 + 2 \sin \theta) = 10[/tex]
Or
[tex]r(1+ \sin \theta) = 5[/tex]
Expand :
[tex]r+ r\sin \theta= 5[/tex]
We now substitute to get:
[tex] \sqrt{ {x}^{2} + {y}^{2} } + y = 5[/tex]
This means that:
[tex]\sqrt{ {x}^{2} + {y}^{2} }=5 - y[/tex]
Square:
[tex] {x}^{2} + {y}^{2} =(5 - y)^{2} [/tex]
Expand:
[tex]{x}^{2} + {y}^{2} =25 - 10y + {y}^{2} [/tex]
[tex]{x}^{2} =25 - 10y [/tex]
[tex]{x}^{2} - 25 = - 10y [/tex]
[tex]y = - \frac{ {x}^{2} }{10} + \frac{5}{2} [/tex]
This is a parabola (0,2.5) and turns upside down.
