3x²+3y²-12x-6y-30=0
first, get rid of all those multiples of 3.
x^2+y^2-4x-2y = 10
x^2-4x+4 + y^2-2y+1 = 10+4+1
(x-2)^2 + (y-1)^2 = 15
To solve the equation 3x² + 3y² - 12x - 6y - 30 = 0, we can follow these steps:
Step 1: Begin by grouping the terms with variables together and the constant term on the other side of the equation:
3x² + 3y² - 12x - 6y = 30
Step 2: Divide both sides of the equation by the common factor of 3 to simplify it:
(3x² - 12x) + (3y² - 6y) = 30
x² - 4x + y² - 2y = 10
Step 3: Complete the square for the x-terms by adding the square of half the coefficient of x to both sides of the equation:
x² - 4x + 4 + y² - 2y = 10 + 4
(x - 2)² + y² - 2y = 14
Step 4: Complete the square for the y-terms by adding the square of half the coefficient of y to both sides of the equation:
(x - 2)² + (y - 1)² - 1 = 14 + 1
(x - 2)² + (y - 1)² = 15
Step 5: Now, the equation is in the form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents the radius. Comparing this to the given equation, we can determine that:
Center of the circle: (2, 1)
Radius of the circle: √15
The given equation is in the general form of a quadratic equation, which describes a conic section called an ellipse. To understand the properties of this ellipse, we need to rewrite the equation in a standard form. The standard form of the equation of an ellipse is:
((x-h)² / a²) + ((y-k)² / b²) = 1
where (h, k) represents the center of the ellipse, and 'a' and 'b' are the lengths of the major and minor axes, respectively.
To convert the given equation into standard form, we complete the square for both the 'x' and 'y' terms.
Starting with the 'x' terms:
3x² - 12x = 0
First, we factor out the common factor of 3:
3(x² - 4x) = 0
Next, we take half of the coefficient of 'x' (which is -4) and square it, then add and subtract the result inside the parenthesis:
3(x² - 4x + 4 - 4) = 0
Simplifying:
3(x² - 4x + 4) - 12 = 0
3(x - 2)² - 12 = 0
Similarly, for the 'y' terms:
3y² - 6y = 0
First, we factor out the common factor of 3:
3(y² - 2y) = 0
Next, we take half of the coefficient of 'y' (which is -2) and square it, then add and subtract the result inside the parenthesis:
3(y² - 2y + 1 - 1) = 0
Simplifying:
3(y² - 2y + 1) - 3 = 0
3(y - 1)² - 3 = 0
Now, we can rewrite the given equation using the completed square forms for 'x' and 'y':
3(x - 2)² - 12 + 3(y - 1)² - 3 - 30 = 0
3(x - 2)² + 3(y - 1)² = 45
Dividing both sides by 45, we get:
((x - 2)² / 3) + ((y - 1)² / 15) = 1
Comparing this equation with the standard form of an ellipse, we can identify that the center of the ellipse is at the point (2, 1), the length of the major axis (2a) is equal to 2√3, and the length of the minor axis (2b) is equal to 2√15. Therefore, 'a' is √3 and 'b' is √15.
To summarize:
- Center of the ellipse: (h, k) = (2, 1)
- Length of the major axis: 2a = 2√3
- Length of the minor axis: 2b = 2√15
By rewriting the equation in standard form and analyzing its properties, we have determined the characteristics of the ellipse.