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# need help

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Find the area of the region enclosed by the graph of $x^2 + y^2 = 2x - 6y + 6 - 18x + 2y$

Oct 28, 2021

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Hello Guest,

$$x^2+y^2=2-x-6y+6-18x+2y$$

$$\mbox {Move the variables to the left side and change their signs}$$

$$x^2+y^2-2x+6y+18x-2y=6$$

$$x^2+y^2+16x+6y-2y=6$$

$$x^2+y^2+16x+4y=6$$

$$\mbox {Use the commutative property to reorder the terms}$$

$$x^2+16x+y^2+4y=6$$

$$\mbox {To complete the square, the same value needs to be added to both sides}$$

$$x^2+16x+?+y^2+4y=6+?$$

$$\mbox {To complete the square } x^2+16x+64=(x+8)^2 \mbox { add 64 to the expression}$$

$$x^2+16x+64+y^2+4y=6+?$$

$$\mbox {Since 64 was added to the left side, also add 64 to the right side}$$

$$x^2+16x+64+y^2+4y=6+64$$

$$\mbox {Use the first binomial } a^2+2ab+b^2=(a+b)^2 \mbox { to factor the expression}$$

$$(x+8)^2+y^2+4y=6+64$$

$$(x+8)^2+y^2+4y=70$$

$$\mbox {To complete the square, the same value needs to be added to both sides}$$

$$(x+8)^2+y^2+4y+?=70+?$$

$$\mbox {To complete the square } y^2+4y+4=(y+2)^2 \mbox { add 4 to the expression}$$

$$(x+8)+y^2+4y+4=70+?$$

$$\mbox {Since 4 was added to the left side, also add 4 to the right side}$$

$$(x+8)+y^2+4y+4=70+4$$

$$\mbox {Use the first binomial } a^2+2ab+b^2=(a+b)^2 \mbox { to factor the expression}$$

$$(x+8)^2+(y+2)^2=70+4$$

$$(x+8)^2+(y+2)^2=74$$

$$\mbox {The equation can be written in the form } (x-p)^2+(y-q)^2=r^2 \mbox { , so it represents a circle with the radius } r=\sqrt{74} \mbox { and the center } (-8,-2)$$

Oct 29, 2021