How do you find the equation of a circle in standard form given #x^2+y^2+4x+6y+4=0#?

Answer 1

Please see the explanation.

The standard Cartesian form of the equation of a circle is:

#(x-h)^2 + (y-k)^2 = r^2" [1]"#

Expand the squares:

#x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2" [2]"#

The given equation is:

#x^2 + y^2 + 4x + 6y + 4 = 0" [3]"#

To make equation [3] look more like equation [2] group the x term together, group the y terms together and move the constant term to the right side:

#x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2" [2]"# #x^2 + 4x + y^2 + 6y = -4" [4]"#
Add #h^2 and k^2# to both sides of equation [4]
#x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2" [2]"# #x^2 + 4x + h^2 + y^2 + 6y + k^2 = h^2 + k^2 -4" [5]"#

To find the value of "h", set the second term of equation [2] equal to the second term of equation [5] and then solve for h:

#-2hx = 4x#
#h = -2#

To find the value of "k", set the fifth term of equation [2] equal to the fifth term of equation [5] and then solve for k:

#-2ky = 6y#
#k = -3#
Now that we know the values of h and k, we know that we can substitute #(x - -2)^2# for the first 3 terms and #(y - -3)^2# for the next 3 terms into equation [5]:
#(x - -2)^2 + (y - -3)^2 = h^2 + k^2 - 4" [6]"#

Substitute -2 for h and -3 for k into equation [6]:

#(x - -2)^2 + (y - -3)^2 = (-2)^2 + (-3)^2 - 4" [7]"#
#(x - -2)^2 + (y - -3)^2 = 4 + 9 - 4" [8]"#

Combine the constants and write the constant as a square:

#(x - -2)^2 + (y - -3)^2 = 3^2" [9]"#

Equation [9] is in standard Cartesian form.

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Answer 2

To find the equation of a circle in standard form given the equation ( x^2 + y^2 + 4x + 6y + 4 = 0 ):

  1. Rearrange the given equation by completing the square for both the ( x ) and ( y ) terms.
  2. Complete the square for the ( x ) terms by adding ((4/2)^2 = 4) inside the parentheses.
  3. Complete the square for the ( y ) terms by adding ((6/2)^2 = 9) inside the parentheses.
  4. Rewrite the equation in standard form by grouping the squared terms together and isolating the constant term on one side.
  5. Write the equation in the standard form of a circle, which is ( (x - h)^2 + (y - k)^2 = r^2 ), where ( (h, k) ) is the center of the circle and ( r ) is the radius.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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