x² + y² + 8x - 4y + 12 = 0
Circle equation: (x - a)² + (y - b)² = r², so,
x² - 2.x.a + a² + y² - 2.y.b + b² - r² = 0
x² + y² - 2ax - 2by + a² + b² - r² = 0
-2ax = 8x
-2a = 8
a = 8/-2
a = -4
-2by = -4y
-2b = -4
b = -4/-2
b = 2
So, (x + 4)² + (y - 2)² - r² = 0
a² + b² - r² = 0
(-4)² + 2² = r²
16 + 4 = r²
r² = 20
So, after all:
(x + 4)² + (y - 2)² = 20
The coordinates of the center is always the opposite of a and b,
So, the center is (-4, 2)
And the radius is r² = 20 => r = √20 => r = 2√5
The coordinates of the center of the circle is: (-4,2)
The length of the radius of the circle is: 2√2
We know that the general equation of a circle i.e. the equation of a circle in square form is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
where (h,k) is the center of the circle and r is the radius of the circle.
The equation of the circle is given by:
[tex]x^2+y^2+8x-4y+12=0[/tex]
Now, on combining the terms of x and y we have:
[tex]x^2+8x+y^2-4y+12=0\\\\i.e.\\\\x^2+2\times 4\times x+y^2-2\times (2)\times y+12=0[/tex]
i.e.
[tex]x^2+4^2-4^2+2\times 4\times x+y^2-(2)^2+(2)^2-2\times 2\times y+12=0\\\\i.e.\\\\x^2+4^2+2\times 4\times x-4^2+y^2-2^2+2^2-4y+12=0\\\\(x+4)^2+(y-2)^2-16-4+12=0\\\\i.e.\\\\(x+4)^2+(y-2)^2-8=0\\\\i.e.\\\\(x+4)^2+(y-2)^2=8[/tex]
Hence, we get:
[tex](x-(-4))^2+(y-2)^2=(2\sqrt{2})^2[/tex]
Hence, the center of the circle is: (-4,2)
and the radius of the circle is: 2√2
