If S and \(S\) ' are two foci of an ellipse \(16 \mathrm{x}^{2}+25 \mathrm{y}^{2}=400\) and PSQ is a focal chord such that \(\mathrm{SP}=16\) then \(\mathrm{S}^{\prime} \mathrm{Q}=\ldots \ldots .\) (a) \((74 / 9)\) (b) \((54 / 9)\) (c) \((64 / 9)\) (d) \((44 / 9)\)

A focal chord of an ellipse is any chord that passes through one of the foci of the ellipse. In the solved problem, we are examining a specific focal chord, PSQ, with the property that its segment SP, leading from one focus S, has a length of 16. The exercise called for determining the length of segment S'Q, from the other focus S' to point Q on the ellipse.

Unfortunately, the provided exercise contained an error in the focal chord condition. It stated that the sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis, which, in our standard form equation, is represented by the variable 2a. For the given ellipse, where the semi-major axis a is 5, this sum is thus 10, not 16 as given in the problem. As a result, there is no such focal chord PSQ where the length PS is 16, since any such chord would contradict the fundamental property of ellipse focal lengths.

The semi-major axis, denoted as 'a', is half of the longest diameter of an ellipse and is a key element in defining the shape of an ellipse. For the ellipse described by the equation in its standard form \(\frac{x^2}{25} + \frac{y^2}{16} = 1\), the semi-major axis is 5. It not only defines the ellipse's width but also features in computing the ellipse’s focal length and in other important properties such as the area of the ellipse, which is given by \(\pi ab\), where 'b' is the semi-minor axis.

Understanding the semi-major axis's role is crucial, as it helps to determine both the geometry of the ellipse and the relationship between points on the ellipse and its foci, which leads to the unique property that the sum of the distances from any point on the ellipse to the foci (S and S') is constant and equal to 2a.

The semi-minor axis of an ellipse, denoted as 'b', is half of the shortest diameter of the ellipse. It is perpendicular to the semi-major axis and defines the ellipse's height. From the standard form equation, we identify the semi-minor axis for our ellipse as 4.

The semi-minor axis, together with the semi-major axis, determines the overall shape of the ellipse. It plays a pivotal role in computing the ellipse's eccentricity, which is a measure of how much the ellipse deviates from being circular. The eccentricity 'e' can be found using the formula \(e = \frac{c}{a}\), where 'c' is the distance from the center to one of the foci, and is calculated as \(c = \sqrt{a^2 - b^2}\). Here, the semi-minor axis forms part of this critical relationship between 'a', 'b', and 'c'.

The ellipse equation in standard form is represented as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where 'a' and 'b' correspond to the lengths of the semi-major and semi-minor axes, respectively. This form of the equation provides a clear and simple way to understand the properties and measurements of an ellipse.

The equation \(\frac{x^2}{25} + \frac{y^2}{16} = 1\) given in the exercise identifies an ellipse centered at the origin with a semi-major axis of 5 units and a semi-minor axis of 4 units. The standard form is the most convenient way to analyze an ellipse as it allows us to easily compute other characteristics such as the location of the foci by using the relationship \(c^2 = a^2 - b^2\) and to ascertain the ellipse's dimensions and orientation. Furthermore, the standard form lays the foundation for exploring the conic sections and their applications in various fields, including physics, engineering, and astronomy.