Convert each equation to its rectangular form.r=−4cosθr=-4\cos\thetar=−4cosθ

( x + 2 ) 2 + y 2 = 4 (x+2)^2+y^2=4 ( x + 2 ) 2 + y 2 = 4

Convert each equation to its rectangular form. r = − 4 cos θ r=-4\cos\theta r = − 4 cos θ

In this problem, we're asked to convert the equation into its rectangular form and the equation that we're given is r equals negative four times the cosine of theta. So let's go ahead and get started with our steps here, starting by trying to get r cosine theta, r sine theta, or r squared. Now here, since I have an r on this left side and I have a cosine theta on that right side, it seems like multiplying both sides by r may be a good choice here. So let's go ahead and do that, multiplying both sides by r. Now on the left, that leaves me with r squared, and that's equal to negative four times r cosine theta. So here I now have an r squared and an r cosine theta just like I wanted. So I can move on to step number two and replace that r cosine theta and r squared using these formulas that I know. So I know that r squared is going to end up giving me x squared plus y squared, and that's equal to negative four x. Now from here, we've replaced that, and we want to rewrite our equation in its standard form. So how are we going to do that here? We have an x and we have a y, and I'm not really sure what to solve for. But looking at our strategies here, I know that if I have x squared plus y squared is equal to some number times x or some number times y, I should go ahead and complete the square. So let's go ahead and do that here. I know that I have this negative four x and I also have this x squared. So I want my x terms to be on the same side and I'm kind of gonna ignore this y squared for a little bit. So if I add four x to both sides, it will cancel on that right side and I have x squared plus four x. And knowing that I'm going to have to complete the square, I'm just going to leave a blank right there and then that y squared is just off to the side and all of this is equal to zero. Now in order to complete the square, remember that we want to get this down to a perfect square and the specific perfect square that I'm looking for here is x plus two squared because I have this four x term. Now what do I need to add to this x squared plus four x in order to make this simplify to this perfect square? Well, I know that if I add four, that will end up giving me that perfect square. But if I add four on that left side, I also need to add it on the right side, so I go ahead and do that. Now I can pull all of this stuff down leaving me with that plus y squared and equals four. Now I have x plus two squared plus y squared equals four and you might recognize this as being the equation of a circle in its standard form. So this is my equation in its rectangular form. Let me know if you have any questions here and thanks for watching.

Convert each equation to its rectangular form. r = − 4 cos ⁡ θ r=-4\cos\theta r = − 4 cos θ

( x + 2 ) 2 + y 2 = 4 (x+2)^2+y^2=4 ( x + 2 ) 2 + y 2 = 4

x 2 + y 2 = 4 x^2+y^2=4 x 2 + y 2 = 4

( x + 2 ) 2 + y 2 = 2 (x+2)^2+y^2=2 ( x + 2 ) 2 + y 2 = 2

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Struggling with Calculus?

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Multiple Choice

Convert each equation to its rectangular form.
$r=-4\cos\theta$

$r=-4x$

$x^2+y^2=4$

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

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

Verified step by step guidance

Step 1: Recall the polar-to-rectangular conversion formulas. The relationships are: x = r * cos(θ) and y = r * sin(θ). Additionally, r² = x² + y².

Step 2: For the equation r = -4 * cos(θ), substitute x = r * cos(θ). This gives r = -4 * (x / r). Multiply through by r to eliminate the denominator, resulting in r² = -4x.

Step 3: For the equation x² + y² = 4, note that this is already in rectangular form. It represents a circle centered at the origin with radius 2.

Step 4: For the equation (x + 2)² + y² = 2, note that this is already in rectangular form. It represents a circle centered at (-2, 0) with radius √2.

Step 5: Verify each conversion by checking the relationships between polar and rectangular coordinates. Ensure that the equations are consistent with the given polar forms.

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