A conic section has the equation x2 + y2 + 10x + 8y = 40. Determine the following: type of conic, domain and range, axes of symmetry, and center. Show your work.
To determine the type of conic section given the equation x^2 + y^2 + 10x + 8y = 40, we can complete the square for both x and y terms separately.
For the x-terms:
1. Move the constant term to the right side: x^2 + 10x + y^2 + 8y = 40
2. Take half of the coefficient of x (10/2 = 5) and square it: (5)^2 = 25
3. Add this result (25) to both sides of the equation: x^2 + 10x + 25 + y^2 + 8y = 40 + 25
4. Simplify: x^2 + 10x + 25 + y^2 + 8y = 65
5. Factor the x terms: (x + 5)^2 + y^2 + 8y = 65
Similarly, for the y-terms:
1. Take half of the coefficient of y (8/2 = 4) and square it: (4)^2 = 16
2. Add this result (16) to both sides of the equation: (x + 5)^2 + y^2 + 8y + 16 = 65 + 16
3. Simplify: (x + 5)^2 + y^2 + 8y + 16 = 81
4. Factor the y terms: (x + 5)^2 + (y + 4)^2 = 81
Now we can determine the type of conic section based on the equation:
Since both the x and y terms are squared and have the same coefficients, we can conclude that the equation represents a circle.
Next, let's determine the domain and range of the conic section:
Since the equation represents a circle and both x and y terms are squared, there are no restrictions on the values of x and y. Therefore, the domain and range are all real numbers.
To find the axes of symmetry:
The circle does not have any specific axes of symmetry since it is symmetrical in all directions.
Finally, let's determine the center of the circle:
From the equation (x + 5)^2 + (y + 4)^2 = 81, we can see that the center of the circle is (-5, -4).
To determine the type of conic, domain, range, axes of symmetry, and center of the given equation x^2 + y^2 + 10x + 8y = 40, we need to manipulate the equation into a standard conic form. Let's go step by step:
Step 1: Group the x-terms and y-terms separately:
(x^2 + 10x) + (y^2 + 8y) = 40
Step 2: Complete the square for x-terms:
To complete the square for x-terms, we take half the coefficient of x (which is 10), square it, and add it to both sides:
(x^2 + 10x + 25) + (y^2 + 8y) = 40 + 25
(x + 5)^2 + (y^2 + 8y) = 65
Step 3: Complete the square for y-terms:
To complete the square for y-terms, we take half the coefficient of y (which is 8), square it, and add it to both sides:
(x + 5)^2 + (y^2 + 8y + 16) = 65 + 16
(x + 5)^2 + (y + 4)^2 = 81
Now we have the equation in the standard form of a conic section: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
Step 4: Comparing with the standard form, we can see that it represents a circle. So, the type of conic is a circle.
Step 5: The domain and range of a circle extend indefinitely in all directions, so the domain and range, in this case, are all real numbers (-∞, +∞).
Step 6: For a circle, the axes of symmetry are the lines passing through the center and parallel to the coordinate axes. In this case, the axes of symmetry are x = -5 (vertical line passing through the center) and y = -4 (horizontal line passing through the center).
Step 7: From the standard form, we can determine the center of the circle by looking at the values of (h, k). In this case, the center is (-5, -4).
To summarize:
- Type of conic: Circle
- Domain: All real numbers (-∞, +∞)
- Range: All real numbers (-∞, +∞)
- Axes of symmetry: Vertical line x = -5 and horizontal line y = -4
- Center: (-5, -4)
x^2 + y^2 + 10x + 8y = 40
x^2+10x + y^2+8y = 40
x^2+10x+25 + y^2+8y+16 = 40+25+16
(x+5)^2 + (y+4)^2 = 81
That help?