The line passing through the extremity \(A\) of the major axis and extremity \(B\) of the minor axis of the ellipse $$ x^{2}+9 y^{2}=9 $$ meets its auxiliary circle at the point \(M .\) Then the area of the triangle with vertices at \(A, M\) and the origin \(O\) is (A) \(\frac{31}{10}\) (B) \(\frac{29}{10}\) (C) \(\frac{21}{10}\) (D) \(\frac{27}{10}\)

The auxiliary circle is a fundamental concept in the geometry of an ellipse. It is defined as the circle that shares the same center as the ellipse and whose radius is equal to the semi-major axis of the ellipse. For any ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the equation of its auxiliary circle is simply \(x^2 + y^2 = a^2\).

The significance of the auxiliary circle lies in its utility for solving various geometric problems related to ellipses, such as finding chord lengths, tangents, and normals to the ellipse. It is also used to derive properties of the ellipse and to simplify calculations, as the circle's symmetry provides a reference that is often easier to work with than the ellipse itself.

Quadratic equations are an integral part of algebra and represent polynomial equations of degree two, with the standard form being \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a \eq 0\). Solving quadratic equations allows us to find the values of \(x\) that make the equation true, and these solutions are called the roots of the equation.

There are several methods to solve a quadratic equation, such as factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The quadratic formula provides a direct way to obtain the roots, and it's extremely useful when equations cannot be easily factored. Understanding how to resolve quadratic equations is essential for tackling a wide range of problems in fields such as physics, engineering, and economics.

Calculating the area of a triangle is a staple in geometry, and there are various methods to do so depending on the information available. The most common formula is \(\frac{1}{2} \times base \times height\), where the base is any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.

In the context of coordinate geometry, if we have the coordinates of the vertices, we can employ determinants or the shoelace formula to find the area. It's important in problems involving triangles to analyze which data points are known so that the most appropriate method for area calculation is used.

The concept of the area of a triangle also extends beyond simple geometry, as it can be applied to calculate the cross product of vectors and find the area spanned by two vectors, making it a versatile mathematical tool.