Question

drag the tiles to the boxes to form correct pairs. not all tiles will be used.
match the circle equations in general form with their corresponding equations in standard form.
2x² + 2y² - 24x - 16y - 8 = 0
5x² + 5y² - 10x + 20y - 30 = 0
3x² + 3y² + 12x + 18y - 15 = 0
x² + y² + 6x - 8y - 10 = 0
x² + y² - 4x + 12y - 20 = 0
x² + y² + 2x - 12y - 9 = 0
(x + 1)² + (y - 6)² = 46
(x + 2)² + (y + 3)² = 18
(x - 2)² + (y + 6)² = 60
(x - 6)² + (y - 4)² = 56

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is \((x - a)^2+(y - b)^2=r^2\), and the general form is \(x^{2}+y^{2}+Dx + Ey+F = 0\) (for equations where the coefficients of \(x^{2}\) and \(y^{2}\) are equal). For non - equal coefficients (but can be made equal by dividing), like \(Ax^{2}+Ay^{2}+Dx + Ey+F = 0\) (\(A
eq0\)).

Step2: Convert general form to standard form (for non - unit coefficients of \(x^{2}\) and \(y^{2}\))

• For \(2x^{2}+2y^{2}-24x - 16y - 8 = 0\), divide by \(2\): \(x^{2}+y^{2}-12x - 8y - 4 = 0\). Complete the square: \((x - 6)^{2}-36+(y - 4)^{2}-16-4 = 0\), \((x - 6)^{2}+(y - 4)^{2}=56\).
• For \(5x^{2}+5y^{2}-10x + 20y - 30 = 0\), divide by \(5\): \(x^{2}+y^{2}-2x + 4y - 6 = 0\). Complete the square: \((x - 1)^{2}-1+(y + 2)^{2}-4-6 = 0\), \((x - 1)^{2}+(y + 2)^{2}=11\) (not in the given standard forms).
• For \(3x^{2}+3y^{2}+12x + 18y - 15 = 0\), divide by \(3\): \(x^{2}+y^{2}+4x + 6y - 5 = 0\). Complete the square: \((x + 2)^{2}-4+(y + 3)^{2}-9-5 = 0\), \((x + 2)^{2}+(y + 3)^{2}=18\).
• For \(x^{2}+y^{2}+6x - 8y - 10 = 0\), complete the square: \((x + 3)^{2}-9+(y - 4)^{2}-16-10 = 0\), \((x + 3)^{2}+(y - 4)^{2}=35\) (not in the given standard forms).
• For \(x^{2}+y^{2}-4x + 12y - 20 = 0\), complete the square: \((x - 2)^{2}-4+(y + 6)^{2}-36-20 = 0\), \((x - 2)^{2}+(y + 6)^{2}=60\).
• For \(x^{2}+y^{2}+2x - 12y - 9 = 0\), complete the square: \((x + 1)^{2}-1+(y - 6)^{2}-36-9 = 0\), \((x + 1)^{2}+(y - 6)^{2}=46\).

Answer:

• \(2x^{2}+2y^{2}-24x - 16y - 8 = 0\) matches \((x - 6)^{2}+(y - 4)^{2}=56\).
• \(3x^{2}+3y^{2}+12x + 18y - 15 = 0\) matches \((x + 2)^{2}+(y + 3)^{2}=18\).
• \(x^{2}+y^{2}-4x + 12y - 20 = 0\) matches \((x - 2)^{2}+(y + 6)^{2}=60\).
• \(x^{2}+y^{2}+2x - 12y - 9 = 0\) matches \((x + 1)^{2}+(y - 6)^{2}=46\).