Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin.
x2 + y2 + x2y2 = 4
A. Symmetric with respect to the x-axis
B. Symmetric with respect to the y-axis
C. Symmetric with respect to the origin
D. Symmetric with respect to the x-axis, the y-axis, and the origin
Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin.
x2 + xy2 + x = 2
A. Symmetric with respect to the x-axis
B. Symmetric with respect to the y-axis
C. Symmetric with respect to the origin
D. Symmetric with respect to the x-axis, the y-axis, and the origin
Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin.
x2 + xy2 + 2y = 1
A. Symmetric with respect to the x-axis
B. Symmetric with respect to the y-axis
C. Symmetric with respect to the origin
D. Not symmetric with respect to the x-axis, the y-axis, or the origin
I choices are :
1. c
2. c
3. d
any help plz
any help????????????
abcds
To test an equation for symmetry with respect to the x-axis, y-axis, and origin, we need to examine if substituting (-x, y), (x, -y), and (-x, -y) into the equation produces the same result as substituting (x, y).
Let's apply this method to each of the given equations:
1. Equation: x^2 + y^2 + x^2y^2 = 4
To test for symmetry with respect to the x-axis, substitute (-x, y) into the equation:
(-x)^2 + y^2 + (-x)^2y^2 = 4
This simplifies to:
x^2 + y^2 + x^2y^2 = 4
Since the result is the same, the equation is symmetric with respect to the x-axis.
To test for symmetry with respect to the y-axis, substitute (x, -y) into the equation:
x^2 + (-y)^2 + x^2(-y)^2 = 4
This simplifies to:
x^2 + y^2 + x^2y^2 = 4
Again, the result is the same, indicating symmetry with respect to the y-axis.
To test for symmetry with respect to the origin, substitute (-x, -y) into the equation:
(-x)^2 + (-y)^2 + (-x)^2(-y)^2 = 4
This simplifies to:
x^2 + y^2 + x^2y^2 = 4
Once more, the result matches, showing symmetry with respect to the origin.
Since the equation is symmetric with respect to the x-axis, y-axis, and origin, the answer is D. Symmetric with respect to the x-axis, the y-axis, and the origin.
2. Equation: x^2 + xy^2 + x = 2
We can follow the same process to test for symmetry:
Substituting (-x, y):
(-x)^2 + y(-y)^2 + (-x) = 2
This simplifies to:
x^2 - xy^2 - x = 2
The result is not the same, indicating the lack of symmetry with respect to the x-axis.
Substituting (x, -y):
x^2 + (-y)(-y)^2 + x = 2
This simplifies to:
x^2 - xy^2 + x = 2
Once again, the result is not the same, indicating the lack of symmetry with respect to the y-axis.
Substituting (-x, -y):
(-x)^2 + (-y)(-y)^2 + (-x) = 2
This simplifies to:
x^2 - xy^2 - x = 2
The result is still not the same, indicating the absence of symmetry with respect to the origin.
Thus, the answer is none of the options. The equation is not symmetric with respect to the x-axis, y-axis, or the origin.
3. Equation: x^2 + xy^2 + 2y = 1
Following the same process as before:
Substituting (-x, y):
(-x)^2 + y(-y)^2 + 2y = 1
This simplifies to:
x^2 - xy^2 + 2y = 1
The result is not the same, indicating the lack of symmetry with respect to the x-axis.
Substituting (x, -y):
x^2 + (-y)(-y)^2 + 2(-y) = 1
This simplifies to:
x^2 - xy^2 - 2y = 1
Once more, the result is not the same, indicating the lack of symmetry with respect to the y-axis.
Substituting (-x, -y):
(-x)^2 + (-y)(-y)^2 + 2(-y) = 1
This simplifies to:
x^2 - xy^2 - 2y = 1
Again, the result is not the same, indicating the absence of symmetry with respect to the origin.
Therefore, the answer is D. Not symmetric with respect to the x-axis, y-axis, or the origin.