Can someone check my answers.
Write the following equations in standard form.
1) x^2 - 2x + y^2 + 4y = 11
answer: (x-1)^2 + (y+2)^2 = 16
2) 4x^2 + 24x - y + 34 = 0
answer: y = 4(x-3)^2 + 2
3) The graph of which equation is a circle?
A. 5x^2 + 10x = 9 + 5y^2
B. 5x^2 - 10x = 0 - 5y^2
C. 5x^2 + 5x + y^2 = 9
answer: C
4)Find the exact solution (s) of each system of equations:
x^2 + y^2 = 25 and 9y = 4x^2
answer: (3,4) (-3,4)
5)Which is the equation of an ellipse with center (-4,2) and a horizontal major axis?
answer: (x+4)^2/16 + (y-2)^2/4 = 1
2 is wrong. Recheck your math regarding the constant.
In 3, C is not a circle because the coefficients of x^2 and y^2 are not equal and not on the same side of the = sign. For a circle, one must to be able to rewrite it as
(x-a)^2 + (y-b)^2 = constant
Would #2 be y=4(x+3)^2-2 and #3 5x^2-10x=0-5y^2?
Yes!
To check your answers for the equations written in standard form, we'll go through each equation and compare it to the given answers.
1) The given equation is: x^2 - 2x + y^2 + 4y = 11
The standard form of a circle equation is: (x - h)^2 + (y - k)^2 = r^2
Comparing the given equation to the standard form, we can see that the answer should be: (x - 1)^2 + (y + 2)^2 = 16
2) The given equation is: 4x^2 + 24x - y + 34 = 0
To convert this equation to the standard form of a parabola, we need to isolate the y term.
Therefore, the answer should be: y = 4(x - 3)^2 + 2
3) To determine which equation represents a circle, we need to compare the given options.
A circle equation in standard form is: (x - h)^2 + (y - k)^2 = r^2
Comparing the equations:
A. 5x^2 + 10x = 9 + 5y^2 -> Not a circle equation
B. 5x^2 - 10x = 0 - 5y^2 -> Not a circle equation
C. 5x^2 + 5x + y^2 = 9 -> This equation matches the standard form of a circle.
Therefore, the answer is: C
4) The given system of equations is:
Equation 1: x^2 + y^2 = 25
Equation 2: 9y = 4x^2
To find the exact solutions, we'll substitute y from Equation 2 into Equation 1:
Substituting y = (4/9)x^2 into Equation 1:
x^2 + (4/9)x^2 = 25
13x^2/9 = 25
13x^2 = 225
x^2 = 225/13
x = ± sqrt(225/13)
Plugging the x-values into Equation 2 to find the corresponding y-values:
For x = sqrt(225/13), y = (4/9)(sqrt(225/13))^2 = 4
Therefore, one solution is (sqrt(225/13), 4).
For x = -sqrt(225/13), y = (4/9)(-sqrt(225/13))^2 = 4
Therefore, another solution is (-sqrt(225/13), 4).
So, the exact solutions are (3, 4) and (-3, 4).
5) The equation of an ellipse with a horizontal major axis and center (h, k) can be written in standard form as:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Comparing this standard form to the given answer:
(x + 4)^2/16 + (y - 2)^2/4 = 1
We can see that the given answer matches the standard form for an ellipse with center (-4, 2) and horizontal major axis.
Therefore, the answer is: (x + 4)^2/16 + (y - 2)^2/4 = 1
I hope this helps! Let me know if you have any other questions.