Understanding Symmetry of Curves

Symmetry is a property of a curve where one part is a mirror image of the rest. We can check for different types of symmetry for a given equation like $xy = 15$.

To check for symmetry about:

• X-axis: Replace $y$ with $-y$ in the equation. If the new equation is the same as the original, the curve is symmetric about the x-axis.
• Y-axis: Replace $x$ with $-x$ in the equation. If the new equation is the same as the original, the curve is symmetric about the y-axis.
• Origin: Replace both $x$ with $-x$ and $y$ with $-y$ in the equation. If the new equation is the same as the original, the curve is symmetric about the origin.
• Line $x=y$: Swap $x$ and $y$ in the equation. If the new equation is the same as the original, the curve is symmetric about the line $x=y$.

Checking Symmetry for the Curve $xy = 15$

Let's apply these checks to the equation $xy = 15$.

Symmetry about X-axis

Original equation: $xy = 15$
Replace $y$ with $-y$: $x(-y) = 15 \implies -xy = 15$

Is $-xy = 15$ the same as $xy = 15$? No, because if $xy=15$, then $-xy$ must be $-15$, and $-15 \neq 15$.

Therefore, the curve $xy = 15$ is not symmetric about the x-axis.

Symmetry about Y-axis

Original equation: $xy = 15$
Replace $x$ with $-x$: $(-x)y = 15 \implies -xy = 15$

Is $-xy = 15$ the same as $xy = 15$? No, for the same reason as above.

Therefore, the curve $xy = 15$ is not symmetric about the y-axis.

Symmetry about the line $x = y$

Original equation: $xy = 15$
Swap $x$ and $y$: $yx = 15$

Is $yx = 15$ the same as $xy = 15$? Yes, because multiplication is commutative ($yx = xy$).

Therefore, the curve $xy = 15$ is symmetric about the line $x = y$.

Symmetry about the Origin

Original equation: $xy = 15$
Replace $x$ with $-x$ and $y$ with $-y$: $(-x)(-y) = 15 \implies xy = 15$

Is $xy = 15$ the same as the original equation? Yes.

Therefore, the curve $xy = 15$ is symmetric about the origin.

Based on the symmetry checks, the curve $xy = 15$ is symmetric about the line $x = y$ and also about the origin. The question asks what the curve is symmetrical about, and one of the options provided is "about the line x = y".

The analysis confirms that the curve $xy=15$ is symmetric about the line $x=y$. This aligns with one of the provided options.

Revision Table: Curve Symmetry

Reviewing how to check symmetry is key for understanding graphs of equations.

• X-axis symmetry: Check if $(x, -y)$ is on the curve when $(x, y)$ is.
• Y-axis symmetry: Check if $(-x, y)$ is on the curve when $(x, y)$ is.
• Origin symmetry: Check if $(-x, -y)$ is on the curve when $(x, y)$ is.
• Line $y=x$ symmetry: Check if $(y, x)$ is on the curve when $(x, y)$ is.
• Line $y=-x$ symmetry: Check if $(-y, -x)$ is on the curve when $(x, y)$ is.

Additional Information: The Curve $xy = 15$

The equation $xy = 15$ represents a standard hyperbola. Specifically, it's a rectangular hyperbola. The general form of such hyperbolas centered at the origin is $xy = c$, where $c$ is a constant. For $xy=15$, $c=15$.

Properties of the hyperbola $xy = c$ ($c \neq 0$):

• The coordinate axes (x-axis and y-axis) are the asymptotes. This means the curve gets infinitely close to the axes but never touches them.
• The curve exists in the first and third quadrants if $c > 0$ (like $xy=15$). It would be in the second and fourth quadrants if $c < 0$.
• It is always symmetric about the origin. This is because if $(x, y)$ satisfies $xy=c$, then $(-x)(-y) = xy = c$ also holds, so $(-x, -y)$ is also on the curve.
• It is always symmetric about the line $y=x$. This is because if $(x, y)$ satisfies $xy=c$, then $yx = c$ also holds, so $(y, x)$ is also on the curve.
• It is also symmetric about the line $y=-x$. If $(x, y)$ satisfies $xy=c$, then $(-y)(-x) = yx = xy = c$ also holds, so $(-y, -x)$ is also on the curve.

So, the hyperbola $xy=15$ has symmetry about the origin, the line $y=x$, and the line $y=-x$. The provided option "about the line x = y" is one of these axes of symmetry.