Question

Find the length of major and minor axis.

Answer 1

To find the length of the major and minor axes of an ellipse, the equation of the ellipse in standard form ₁ can be utilized, where ₂ and ₃ represent the lengths of the major and minor axes, respectively.

Step-by-step explanation:

The equation of an ellipse in standard form is x²/a² + y²/b² = 1, where a represents the length of the major axis ₂, and b represents the length of the minor axis₃. To determine these values, one must identify the coefficients of x² and y² in the equation of the ellipse. If the equation of the ellipse is not already in standard form, it can be manipulated algebraically to get it into the standard form equation.

For example, consider an equation of an ellipse like x²/16 + y²/9 = 1. From this equation, a² is 16, so the length of the major axis ₂ is
`\(2a = 2√(16) = 8\)`. Similarly, b² is 9, making the length of the minor axis ₃ equal to
`\(2b = 2√(9) = 6\)`. Therefore, the major axis has a length of 8 units, and the minor axis has a length of 6 units.

Understanding the standard form equation of an ellipse and identifying the coefficients of x² and y² allows for the determination of the major and minor axes lengths. This knowledge is crucial in geometry and engineering applications where the properties of ellipses are relevant, aiding in accurate calculations and design considerations.

Answer 2

The major and minor axes of an ellipse are the longest and shortest diameters, respectively. The semi-major axis 'a' is half of the major axis and can be calculated using aphelion and perihelion distances. The full lengths of the axes are twice their corresponding semi-axes, with the major axis being '2a' and the minor axis being '2b'.

Step-by-step explanation:

The question pertains to finding the lengths of the major and minor axes of an ellipse, which are key concepts in the study of conic sections in mathematics. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest diameter.

The lengths of the major and minor axes can be obtained from the lengths of their corresponding semi-axes. The semi-major axis, denoted by 'a', is half the length of the major axis, while the semi-minor axis, denoted by 'b', is half the length of the minor axis.

The length of the semi-major axis can also be calculated if one knows the aphelion and perihelion distances of an object's elliptical orbit by using the formula a = ½(ra + rp), where 'ra' is the aphelion distance and 'rp' is the perihelion distance. Once the semi-major axis 'a' is known, the major axis is simply twice that length, or 2a.

Similarly, to find the semi-minor axis 'b', additional orbital parameters or geometrical properties of the ellipse would be needed, after which the minor axis is found as 2b.

To clearly depict these elements in a diagram, one would show an ellipse with the major and minor axes labeled, and potentially also the foci if the context requires understanding the elliptical properties related to Kepler's laws or orbital mechanics.