1) First of all, the first equation is the eq. of a circle, whose the general form is:
(x-a)^2+ (y-b)^2 - R^2=0, where R is the circle radius and a,b are the coordinates of the circle center.
E(x)=(6x)^2-12x+(6y)^2+36y=36
First of all we'll have the common factor with value=36
(6x)^2-12x+(6y)^2+36y=36
36*(x^2 - (1/3)*x+ y^2 + y]=36
We'll simplify with the value 36
x^2 - (1/3)*x+ y^2 + y=1
We'll try to emphasize that x^2 - (1/3)*x is a part from the developing: x^2 - 2*(1/2)*(1/3)*x + (1/2)^2= [x-(1/2)]^2
We have observed that we've added the term (1/2)^2, so we have to get rid of it.
[x-(1/2)]^2-(1/2)^2
Same way we'll do with the second part of the expression
y^2 + y=y^2 + 2*(1/2)*y+(1/2)^2 - (1/2)^2
y^2 + y=[y+(1/2)]^2 - (1/2)^2
The expression will become:
E(x)=(x - 1/6)^2 + (y+ 1/2)^2- [sqrt(46/36)]^2=0
This is the equation of a circle, which has the center
C(1/6, -1/2) and R^2=46/36
2) We'll try again to form the eq. of a circle, in the same way we've did at the first example.
(4y)^2-8y+(9x)^2-54x=49
(4y)^2-8y=(4y)^2-2*4y +1-1=(4y-1)^2 - 1
(9x)^2-54x=(9x)^2-2*9x*3 + 9-9=(9x-3)^2-9
(9x-3)^2-9 + (4y-1)^2 - 1 -9 - 49=0
(9x-3)^2-9 + (4y-1)^2 - 59=0
This is the equation of a circle, which has the center
C(3/9, 1/4)=C(1/3, 1/4) and R^2=59
3)We'll try again to form the eq. of a circle, in the same way we've did at the first example.
E(x)=9y^2+108y+4x^2-56x=-484
If we'll consider just the unknown y and the unknown x, raised to the 2nd power,than we'll have
9y^2+108y=(3y)^2-2*3y*18+18^2 -18^2=(3y+18)^2-18^2
4x^2-56x=(2x)^2-2*2x*14+14^2 - 14^2=(2x-14)^2-14^2
E(x)=(3y+18)^2-18^2+(2x-14)^2-14^2+484=0
(3y+18)^2+(2x-14)^2-36=0
This is the equation of a circle, which has the center
C(-18/3, 14/2)=C(-6, 7) and R^2=36, R=6
No comments:
Post a Comment