91Ó°ÊÓ

In an ellipse, the foci are two special points: any point on the ellipse is such that the sum of its distances from the two foci is constant. This distinct property is what gives the ellipse its unique shape. To locate the foci in an ellipse given by the equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) (assuming \( b > a \), meaning the ellipse is vertical), you use the formula \( c = \sqrt{b^{2} - a^{2}} \). Here, \( c \) is the distance from the center of the ellipse to each focus, measured along the major axis.
For our example \( \frac{x^{2}}{49} + \frac{y^{2}}{81} = 1 \), we found \( a = 7 \) and \( b = 9 \). Calculating \( c \), we get \( c = \sqrt{81 - 49} = \sqrt{32} \). The foci are then located symmetrically on the major axis at points \( (0, \sqrt{32}) \) and \( (0, -\sqrt{32}) \). By placing them on the graph, we help visualize this fundamental aspect of ellipses.

The semi-major axis of an ellipse is the longest radius that extends from the center to the edge of the ellipse. Understanding this concept is crucial, as it defines the size and orientation of the ellipse. In an equation of the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), if \( b > a \), then \( b \) is the semi-major axis.
For our particular equation, \( \frac{x^{2}}{49} + \frac{y^{2}}{81} = 1 \), since \( b = 9 \) is greater than \( a = 7 \), \( b \) signifies the semi-major axis. The length of the semi-major axis is \( 9 \).
This axis is vertical, indicating the ellipse is oriented upwards and downwards along the y-axis. When drawing the ellipse, measuring \( 9 \) units from the center in both the positive and negative directions on the y-axis accurately locates the extremities of this axis.

The semi-minor axis is the shorter radius of an ellipse that extends from the center to the edge along the axis perpendicular to the semi-major. It's vital for shaping the ellipse's form. In the equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), assuming \( b > a \) for a vertical ellipse, \( a \) represents the semi-minor axis.
In our example \( \frac{x^{2}}{49} + \frac{y^{2}}{81} = 1 \), \( a = 7 \). This value is the length of the semi-minor axis, making it 7 units.
This axis runs horizontally and helps determine the width of the ellipse. On the graph, the semi-minor axis extends 7 units in both the positive and negative directions on the x-axis. Understanding both the major and minor axes helps in accurately sketching the ellipse and understanding its proportions and location.