Graphing an Ellipse with Center at the Origin

Graph the equation: \(9x^{2}+4y^{2}=36\)

Solution  To put the equation in standard form, we divide each side by \(36\). \[ \dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1 \]

The graph of this equation is an ellipse. Since the larger number is under \( y^{2}\), the major axis is along the \(y\)-axis. The center is at the origin (0, 0). It is easiest to graph an ellipse by finding its intercepts: \[ \begin{array}{rcl@{\qquad\qquad\qquad\qquad}rcl} {x\hbox{-intercepts: Let }y=0} & {y\hbox{-intercepts: Let }x=0} \\[5pt] \dfrac{x^{2}}{4}+\dfrac{0^{2}}{9} &=& 1 & 0^{2}+\dfrac{y^{2}}{9}&=&1 \\[9pt] x^{2}&=&4 & y^{2}&=&9 \\[5pt] {x=-2 \hbox{ and } x=2} & {y=-3 \hbox{ and } y=3} \end{array} \]

A-24

The points \(( -2,0) \), \(( 2,0) \), \(( 0,-3) \) , \(( 0,3)\) are the intercepts of the ellipse. See Figure 40.