The graph of an equation of the form $A x ^ { 2 } + B y ^ { 2 } = C$ , where A , B , and C are nonzero real numbers and $A \neq B$ is an hyperbola.

Question

Quizplus · Accepted Answer

The equation of an hyperbola in standard form is usually given as $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1$, where (h,k) is the center of the hyperbola and a and b are the length of the semi-axes. However, by rearranging the terms, we can obtain the equation $A x ^ { 2 } + B y ^ { 2 } = C$ , where A , B , and C are nonzero real numbers and $A \neq B$. This is the equation of a hyperbola with its center at the origin and its transverse axis along the x-axis or y-axis, depending on whether A or B is the larger coefficient. Therefore, the statement is true.

The Graph of an Equation of the Form Ax2+By2=CA x ^ { 2 } + B y ^ { 2 } = CAx2+By2=C

The Graph of an Equation of the Form $A x ^ { 2 } + B y ^ { 2 } = C$