Final answer:
To determine the endpoints of the minor and major axes of an ellipse, one must have the standard form of the ellipse equation. The endpoints are derivable by adding and subtracting the semi-major (a) and semi-minor (b) axis lengths from the center coordinates of the ellipse. An accurate graph or equation is necessary to perform this correctly.
Step-by-step explanation:
The question asks to find the endpoints of the minor and major axis for an ellipse, which in standard form is represented as \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\). However, the equation provided in the question appears to be incomplete. It should have the forms \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\) for a horizontal major axis, or \((x-h)^2/b^2 + (y-k)^2/a^2 = 1\) for a vertical major axis, where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. Based on Figure 7.6, the semi-major axis is half of the major axis, and the semi-minor axis is half of the minor axis. The endpoints of these axes can be found by adding and subtracting \(a\) and \(b\) to/from the coordinates of the ellipse's center \((h, k)\).
To find the endpoints accurately, we need a complete, correct equation. Assuming a correct equation, the endpoints of the axes are found as follows:
- For the major axis: the endpoints are \((h+a, k)\) and \((h-a, k)\) if the major axis is horizontal or \((h, k+a)\) and \((h, k-a)\) if it is vertical.
- For the minor axis: the endpoints are \((h, k+b)\) and \((h, k-b)\) if the minor axis is perpendicular to a horizontal major axis or \((h+b, k)\) and \((h-b, k)\) if it is perpendicular to a vertical major axis.
Note: To use this method correctly, the provided ellipse equation must be in the standard form and properly identified as having either a horizontal or vertical major axis. If the graph of the ellipse shows that the longer diameter runs left and right, the ellipse has a horizontal major axis; if it runs up and down, then the major axis is vertical. The lengths \(a\) and \(b\) can be found by carefully examining the coefficients of the corresponding terms in the equation.