Test algebraically to determine whether the equations graph

Test algebraically to determine whether the equation\'s graph is symmetric with respect to the X-axis, Y-axis, or origin. x^2 - y^2 = 81 Symmetric with respect to the origin. Symmetric with respect to the X-axis, Y-axis, and origin. No symmetry Symmetric with respect to the X-axis. Symmetric with respect to the Y-axis.

Solution

Given equation is X² - Y² = 81.

Checking whether the graph is symmetric about X-axis:

If Y= -Y in the equation results in the original equation then the graph is symmetric is about X-axis.

X² - Y² = 81.

X² - (-Y)² = 81.

X² - Y² = 81.

Therefore, the given equation\'s graph is symmetric about X-axis.

Checking whether the graph is symmetric about Y-axis:

If X= -X in the equation results in the original equation then the graph is symmetric is about Y-axis.

X² - Y² = 81.

(-X)² - Y² = 81.

X² - Y² = 81.

Therefore, the given equation\'s graph is symmetric about Y-axis.

Checking whether the graph is symmetric about origin:

If X=-X and Y= -Y in the equation results in the original equation then the graph is symmetric is about origin.

X² - Y² = 81.

(-X)² - (-Y)² = 81.

X² - Y² = 81.

Therefore, the given equation\'s graph is symmetric about origin.

So, second option is correct which is that the graph is symmetric about X-axis, Y-axis and origin.