Convert the following equation into the standard form of the equation of a circle.
x^2 + y^2 + 6x – 4y – 3 = 0
To convert the equation into standard form, we need to complete the square. Assume that an expression is given:
aX^2+ bX
To complete the square, we can add and subtract \left(\dfrac{b}{2a}\right)^2. So we have:
x^2 + y^2 + 6x – 4y – 3 = 0
\left[x^2 + 6x +\left(\dfrac{6}{2}\right)^2\right]- \left(\dfrac{6}{2}\right)^2 + \left[y^2 – 4y +\left(\dfrac{4}{2}\right)^2\right]- \left(\dfrac{4}{2}\right)^2 -3=0
\left[x^2 + 6x +\left(\dfrac{6}{2}\right)^2\right]- \left(\dfrac{6}{2}\right)^2 + \left[y^2 – 4y +\left(\dfrac{4}{2}\right)^2\right]- \left(\dfrac{4}{2}\right)^2 -3=0
\left[x^2 + 6x +9\right]- 9 + \left[y^2 – 4y + 4\right]- 4 -3=0
\left(x+3\right)^2+\left(y-2\right)^2 =16
The standard form of this equation is \left(x+3\right)^2+\left(y-2\right)^2=16
Convert the following equation into the general form of the equation of a circle.
\left(x+2\right)^2 + \left(y+2\right)^2 = 1
To convert the equation into general form, we need to extend and simplify the standard form:
\left(x+2\right)^2 + \left(y+2\right)^2 = 1
x^2+4x+4+ y^2 +4y+4=1
x^2+y^2 + 4x+4y+ 4+4-1=0
x^2+y^2 + 4x+ 4y+7=0
The general form of this equation is x^2+y^2 + 4x+ 4y+7=0
Convert the following equation into the standard form of the equation of a circle.
x^2 + y^2 -10x – 8y +40 = 0
To convert the equation into standard form, we need to complete the square. Assume that an expression is given:
aX^2+ bX
To complete the square, we can add and subtract \left(\dfrac{b}{2a}\right)^2. Thus:
x^2 + y^2 -10x – 8y +40 = 0
\left[x^2 -10x +\left(\dfrac{10}{2}\right)^2\right]- \left(\dfrac{10}{2}\right)^2 + \left[y^2 – 8y +\left(\dfrac{8}{2}\right)^2\right]- \left(\dfrac{8}{2}\right)^2+40=0
\left[x^2 -10x +\left(\dfrac{10}{2}\right)^2\right]- \left(\dfrac{10}{2}\right)^2 + \left[y^2 –8y +\left(\dfrac{8}{2}\right)^2\right]- \left(\dfrac{8}{2}\right)^2 +40=0
\left[x^2-10x +25\right]-25 + \left[y^2 – 8y +16\right]- 16+40=0
\left(x-5\right)^2+\left(y-4\right)^2 =1
The standard form of this equation is \left(x-5\right)^2+\left(y-4\right)^2=1
Convert the following equation into the general form of the equation of a circle.
\left(x-3\right)^2 + \left(y-1\right)^2 = 4
To convert the equation into general form, we need to extend and simplify the standard form:
\left(x-3\right)^2 + \left(y-1\right)^2 = 4
\left(x^2-6x+9\right)+\left(y^2-2y+1\right) = 4
x^2-6x+9+y^2-2y+1 = 4
x^2+y^2- 6x-2y+6=0
The general form of this equation is x^2+y^2- 6x-2y+6=0
Convert the following equation into the general form of the equation of a circle.
\left(x\right)^2 + \left(y+3\right)^2 = 2
To convert the equation into general form, we need to extend and simplify the standard form:
\left(x\right)^2 + \left(y+3\right)^2 = 2
\left(x^2\right)+\left(y^2+6y+9\right) = 2
x^2+y^2+6y+9 = 2
x^2+y^2+6y+7=0
The general form of this equation is x^2+y^2+6y+7=0
Convert the following equation into the general form of the equation of a circle.
\left(x+3\right)^2 + \left(y\right)^2 = 4
To convert the equation into general form, we need to extend and simplify the standard form:
\left(x+3\right)^2 + \left(y\right)^2 = 4
\left(x^2+6x+9\right)+\left(y^2\right) = 4
x^2+6x+9+y^2 = 4
x^2+y^2+6x+5=0
The general form of this equation is x^2+y^2+6x+5=0
Convert the following equation into the general form of the equation of a circle.
\left(x-4\right)^2 + \left(y+4\right)^2 = 4
To convert the equation into general form, we need to extend and simplify the standard form:
\left(x-4\right)^2 + \left(y+4\right)^2 = 4
\left(x^2-8x+16\right)+\left(y^2+8y+16\right) = 4
x^2-8x+16+y^2+8y+16 = 4
x^2+y^2- 8x+8y+28=0
The general form of this equation is x^2+y^2- 8x+8y+28=0