Mathematics

Problem

The line passing through the extremity

A

of the major axis and extremity

B

of the minor axis of the ellipse

x^2 + 9y^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}

\frac{29}{10}

\frac{21}{10}

\frac{27}{10}

Intuitive explanation

The key is recognizing that the ellipse $x^2 + 9y^2 = 9$ has a specific major axis, minor axis, and an associated auxiliary circle (which has radius equal to the semi-major axis). You need to find where the line through two specific points on the ellipse intersects this larger circle, then calculate the triangle's area using the distance formula and basic geometry. The auxiliary circle is often the "hidden" tool that makes these problems elegant—it's larger than the ellipse but shares the same center.

Intuitive explanation

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