how an I tell if this is a circle or an ellipse? also, how am I supposed to solve it?
-x^2+2y^2+8x+3=0
I answered a similar question for Jordan last night at 11:20
divide your equation by -1 to make it start with a positive x^2 term.
At this point you should realize that it is neither a circle nor an ellipse, but rather a hyperbola.
Follow the same steps.
put it in standard form.
for the x, complete the square
(x-k)^2/a^2 + (y-h)^2/b^2 =1
if a an b are equal, it is a circle
To determine whether the equation -x^2 + 2y^2 + 8x + 3 = 0 represents a circle or an ellipse, we need to manipulate the equation into standard form and examine its coefficients. Here's the step-by-step solution:
1. Divide the equation by -1 to make the leading coefficient of x^2 positive:
x^2 - 2y^2 - 8x - 3 = 0
2. Reorder the equation to group the x and y terms together:
x^2 - 8x - 2y^2 - 3 = 0
3. Complete the square for the x terms by adding and subtracting the square of half the coefficient of x, which is (8/2)^2 = 16:
x^2 - 8x + 16 - 16 - 2y^2 - 3 = 0
4. Simplify the equation:
(x - 4)^2 - 19 - 2y^2 = 0
5. Rearrange the equation to fit the standard form:
(x - 4)^2 - 2y^2 = 19
6. Divide each term by 19 to make the right side equal to 1:
((x - 4)^2)/19 - (2y^2)/19 = 1
By comparing the equation to the standard form equation of a conic section, we can determine that it represents a hyperbola and not a circle or an ellipse.
To determine whether a conic section is a circle or an ellipse, we need to further analyze the standard form equation. In the standard form equation for an ellipse, both the x and y terms should have the same denominator for their squared variables. If the denominators are different, as is the case here, it indicates a hyperbola.
Therefore, the given equation represents a hyperbola.
Follow the same steps explained above to put the equation in standard form for x, complete the square, and compare the denominators for x^2 and y^2 to determine the conic section. If the denominators are the same, it would represent a circle.