The length of the diameter of the circle which the equation x + y + 18x-22y = -153 is 3.5 units long.
The equation of the circle is the equation which is used to represent the circle in the algebraic equation form with the value of center point in the coordinate plane and measure of radius.
The standard form of the equation of the circle can be given as,
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Here (h,k) is the center of the circle and (r) is the radius of the circle.
The equation of the circle given as,
[tex]x^2 + y^2 + 18x-22y = -153[/tex]
Rearrange the equation to make it as standard form,
[tex]x^2 + y^2 + 18x-22y = -153\\x^2 + 18x+ y^2-22y = -153[/tex]
Add 121 and 81 both side of the equation,
[tex]x^2 + 18x+81+ y^2-22y+121 = -153+81+121\\x^2 + 18x+9^2+ y^2-22y+11^2 =41[/tex]
Using the formula of whole square,
[tex](x+9)^2+(x-11)^2=7^2\\(x-(-9))^2+(x-11)^2=7^2[/tex]
Compare it with standard equation, we get the radius of the circle r is 7.
As the radius is half of the diameter. Thus, the diameter of the circle is,
[tex]d=\dfrac{7}{2}\\d=3.5\rm\; units[/tex]
Thus, the length of the diameter of the circle which the equation x + y + 18x-22y = -153 is 3.5 units long.
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