How to find the center and radius of the circle x^2+y^2+4x-8y+4=0?

Franco Avila

Franco Avila

Answered question

2023-03-11

How to find the center and radius of the circle x 2 + y 2 + 4 x - 8 y + 4 = 0 ?

Answer & Explanation

loise9oa

loise9oa

Beginner2023-03-12Added 4 answers

The general form of the equation of a circle is
a a x 2 + y 2 + 2 g x + 2 f y + c = 0 a a ̲ ¯
with centre = a a ( - g - f ) a a ̲ ¯
and radius = a a g 2 + f 2 - c a a ̲ ¯
x 2 + y 2 + 4 x - 8 y + 4 = 0 is in this form
We must first determine g, f, and c in order to determine the center and radius.
By comparing the coefficients of 'like terms' in the given equation with the general form.
2g = 4 → g = 2 , 2f = -8 → f = -4 and c = 4
centre = ( - g , - f ) = ( - 2 , 4 )
and radius = 2 2 + ( - 4 ) 2 - 4 = 4 + 16 - 4 = 4
Alternatively Use the method of completing the square
Add ( 1 2 coefficient of x/y terms ) 2 to both sides
( x 2 + 4 x +4 ) + ( y 2 - 8 y +16 ) = 4+16 - 4
( x + 2 ) 2 + ( y - 4 ) 2 = 16
The standard form of the equation of a circle is
a a ( x - a ) 2 + ( y - b ) 2 = r 2 a a ̲ ¯
where (a ,b) are the coordinates of the centre and r, the radius.
By comparison of equation with standard form.
a = -2 , b = 4 and r = 4
Thus centre = (-2 ,4) and radius = 4

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