#1 6xto the 2nd power-12+6yto the 2nd power+36y=36 #2 4y2ndpower-8y+9x2nd power-54x=49 #3 9y2nd power+108y+4x2ndpower-56x=-484hey need help with these ellipses

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

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