Find the area of the region enclosed by the graph of $x^2 + y^2 = 2x - 6y + 6 - 18x + 2y$
+678 Hello Guest,
\(x^2+y^2=2-x-6y+6-18x+2y\)
\(\mbox {Move the variables to the left side and change their signs}\)
\(x^2+y^2-2x+6y+18x-2y=6\)
\(x^2+y^2+16x+6y-2y=6\)
\(x^2+y^2+16x+4y=6\)
\(\mbox {Use the commutative property to reorder the terms}\)
\(x^2+16x+y^2+4y=6\)
\(\mbox {To complete the square, the same value needs to be added to both sides}\)
\(x^2+16x+?+y^2+4y=6+?\)
\(\mbox {To complete the square } x^2+16x+64=(x+8)^2 \mbox { add 64 to the expression}\)
\(x^2+16x+64+y^2+4y=6+?\)
\(\mbox {Since 64 was added to the left side, also add 64 to the right side}\)
\(x^2+16x+64+y^2+4y=6+64\)
\(\mbox {Use the first binomial } a^2+2ab+b^2=(a+b)^2 \mbox { to factor the expression}\)
\((x+8)^2+y^2+4y=6+64\)
\((x+8)^2+y^2+4y=70\)
\(\mbox {To complete the square, the same value needs to be added to both sides}\)
\((x+8)^2+y^2+4y+?=70+?\)
\(\mbox {To complete the square } y^2+4y+4=(y+2)^2 \mbox { add 4 to the expression}\)
\((x+8)+y^2+4y+4=70+?\)
\(\mbox {Since 4 was added to the left side, also add 4 to the right side}\)
\((x+8)+y^2+4y+4=70+4\)
\(\mbox {Use the first binomial } a^2+2ab+b^2=(a+b)^2 \mbox { to factor the expression}\)
\((x+8)^2+(y+2)^2=70+4\)
\((x+8)^2+(y+2)^2=74\)
\(\mbox {The equation can be written in the form } (x-p)^2+(y-q)^2=r^2 \mbox { , so it represents a circle with the radius } r=\sqrt{74} \mbox { and the center } (-8,-2)\)
