Question

Convert the polar equation to rectangular form. and Convert the rectangular equation to polar form

please show or explain the steps of work so I can learn to do this myself

Answer 1

Answer:

4) $(x-1)^2 + (y - 1)^2 = 2$

Step-by-step explanation:

4) We can convert this polar equation to rectangular form using the following formulas:

• $r^2 = x^2 + y^2$
• $r \sin(\theta) = y$
• $r\cos(\theta) = x$
• $\tan(\theta) = \dfrac{y}{x}$

We need to manipulate the equation so some of the terms match these formulas.

$r=2\cos(\theta) + 2\sin(\theta)$

↓ dividing both sides by $2$

$r / 2=\cos(\theta) + \sin(\theta)$

↓ multiplying both sides by $r$

$\dfrac{r^2}{2}=r\cos(\theta) + r\sin(\theta)$

Now, we can apply the polar conversion formulas to get the equation:

$\dfrac{x^2 + y^2}{2}=x + y$

This can be manipulated to get a circle standard form equation.

↓ multiplying both sides by 2

$x^2 + y^2=2x + 2y$

↓ completing the square for $x$

    ↓ moving the x's to one side

    $x^2 -2x=-y^2 + 2y$

    ↓ adding (-2/2)², or 1, to both sides

    $x^2 -2x + 1=-y^2 + 2y + 1$

    ↓ factoring the perfect square

    $(x-1)^2=-y^2 + 2y + 1$

↓ completing the square for $y$

    ↓ factoring a (-1) out of the right side

    $(x-1)^2=-1(y^2 - 2y - 1)$

    ↓ adding (-1)(+2) to both sides

    $(x-1)^2 - 2=-1(y^2 - 2y - 1 + 2)$

    ↓ simplifying

    $(x-1)^2 - 2=-1(y^2 - 2y + 1)$

    ↓ factoring the perfect square

    $(x-1)^2 - 2=-(y - 1)^2$

↓ adding $(y-1)^2$ to both sides

$(x-1)^2 + (y - 1)^2 - 2 = 0$

↓ adding 2 to both sides

$\boxed{(x-1)^2 + (y - 1)^2 = 2}$