Answer :
To convert the general form of the equation of a circle \( x^2 + y^2 + 8x + 22y + 37 = 0 \) to its standard form, we complete the square for the \( x \) and \( y \) terms.
1. Completing the square for the \( x \)-terms:
[tex]\[ x^2 + 8x \][/tex]
Take the coefficient of \( x \), which is 8, divide it by 2 to get 4, and square it to get 16:
[tex]\[ (x + 4)^2 - 16 \][/tex]
2. Completing the square for the \( y \)-terms:
[tex]\[ y^2 + 22y \][/tex]
Take the coefficient of \( y \), which is 22, divide it by 2 to get 11, and square it to get 121:
[tex]\[ (y + 11)^2 - 121 \][/tex]
3. Adjusting the constant term:
Rewrite the original equation with the completed squares and balance the constants accordingly:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 \][/tex]
Simplify the constants:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
So, the equation in standard form is:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
The center of the circle is at the point \( (-4, -11) \).
Therefore:
- The first blank should be filled with \( 4 \).
- The second blank should be filled with \( 11 \).
- The third blank should be filled with \( 100 \).
- The fourth and fifth blanks should be filled with \( -4 \) and \( -11 \) respectively.
So the fully filled formula would look like this:
The general form of the equation of a circle is \( x^2 + y^2 + 8x + 22y + 37 = 0 \).
The equation of this circle in standard form is \( (x + 4)^2 + (y + 11)^2 = 100 \).
The center of the circle is at the point [tex]\( (-4, -11) \)[/tex].
1. Completing the square for the \( x \)-terms:
[tex]\[ x^2 + 8x \][/tex]
Take the coefficient of \( x \), which is 8, divide it by 2 to get 4, and square it to get 16:
[tex]\[ (x + 4)^2 - 16 \][/tex]
2. Completing the square for the \( y \)-terms:
[tex]\[ y^2 + 22y \][/tex]
Take the coefficient of \( y \), which is 22, divide it by 2 to get 11, and square it to get 121:
[tex]\[ (y + 11)^2 - 121 \][/tex]
3. Adjusting the constant term:
Rewrite the original equation with the completed squares and balance the constants accordingly:
[tex]\[ x^2 + y^2 + 8x + 22y + 37 = (x + 4)^2 - 16 + (y + 11)^2 - 121 + 37 \][/tex]
Simplify the constants:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
So, the equation in standard form is:
[tex]\[ (x + 4)^2 + (y + 11)^2 = 100 \][/tex]
The center of the circle is at the point \( (-4, -11) \).
Therefore:
- The first blank should be filled with \( 4 \).
- The second blank should be filled with \( 11 \).
- The third blank should be filled with \( 100 \).
- The fourth and fifth blanks should be filled with \( -4 \) and \( -11 \) respectively.
So the fully filled formula would look like this:
The general form of the equation of a circle is \( x^2 + y^2 + 8x + 22y + 37 = 0 \).
The equation of this circle in standard form is \( (x + 4)^2 + (y + 11)^2 = 100 \).
The center of the circle is at the point [tex]\( (-4, -11) \)[/tex].
