Question
lines of symmetry for 4x^2 + y^2 = 9
Answers
GPT-4o mini
To find the lines of symmetry for the equation \( 4x^2 + y^2 = 9 \), we first recognize that this is the equation of an ellipse.
To rewrite this equation in standard form, we can divide the entire equation by 9:
\[
\frac{4x^2}{9} + \frac{y^2}{9} = 1
\]
This can be further simplified to:
\[
\frac{x^2}{\frac{9}{4}} + \frac{y^2}{9} = 1
\]
From here, we can identify that the ellipse is centered at the origin \((0, 0)\) with a semi-major axis length of \(3\) (along the \(y\)-axis) and a semi-minor axis length of \(\frac{3}{2}\) (along the \(x\)-axis).
**Lines of Symmetry:**
1. **Vertical Axis**: The line \(x = 0\) (the y-axis) is a line of symmetry.
2. **Horizontal Axis**: The line \(y = 0\) (the x-axis) is also a line of symmetry.
3. **Diagonal Axes**: The slopes of the diagonals (line equations of the form \(y = mx\)) do not apply since this ellipse is aligned with the coordinate axes.
Thus, the only lines of symmetry for the ellipse described by the equation \(4x^2 + y^2 = 9\) are the x-axis and the y-axis.
In summary, the lines of symmetry are:
- \(y = 0\) (the x-axis)
- \(x = 0\) (the y-axis)
To rewrite this equation in standard form, we can divide the entire equation by 9:
\[
\frac{4x^2}{9} + \frac{y^2}{9} = 1
\]
This can be further simplified to:
\[
\frac{x^2}{\frac{9}{4}} + \frac{y^2}{9} = 1
\]
From here, we can identify that the ellipse is centered at the origin \((0, 0)\) with a semi-major axis length of \(3\) (along the \(y\)-axis) and a semi-minor axis length of \(\frac{3}{2}\) (along the \(x\)-axis).
**Lines of Symmetry:**
1. **Vertical Axis**: The line \(x = 0\) (the y-axis) is a line of symmetry.
2. **Horizontal Axis**: The line \(y = 0\) (the x-axis) is also a line of symmetry.
3. **Diagonal Axes**: The slopes of the diagonals (line equations of the form \(y = mx\)) do not apply since this ellipse is aligned with the coordinate axes.
Thus, the only lines of symmetry for the ellipse described by the equation \(4x^2 + y^2 = 9\) are the x-axis and the y-axis.
In summary, the lines of symmetry are:
- \(y = 0\) (the x-axis)
- \(x = 0\) (the y-axis)