If the conic is an ellipse and the origin lies outside the...

If the conic is an ellipse and the origin lies outside the ellipse, then
(a) $0<\mathrm{e}<\frac{\sqrt{17}}{9}$
(b) $\frac{4}{\sqrt{17}}<\mathrm{e}<1$
(c) $\frac{\sqrt{17}}{9}<\mathrm{e}<1$
(d) $\frac{\sqrt{17}}{9}<\mathrm{e}<\frac{4}{\sqrt{17}}$

Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of all points in the plane for which the sum of the distances to two fixed points (the foci) is constant. It is characterized by its major and minor axes, center, and foci, and is represented by a standard equation that relates these elements. Understanding the ellipse is fundamental in geometry and appears in various applications such as orbital mechanics and optics.

Eccentricity

Eccentricity is a parameter that quantifies the deviation of a conic section from a circle. For an ellipse, its value ranges from 0 to 1, with values closer to 0 indicating a shape that is nearly circular and values closer to 1 indicating a more elongated shape. This concept is critical in determining the shape of an ellipse and comparing it with other conic sections, as it directly influences the relationship between the distances of any point on the ellipse to its foci.

Position of a Point Relative to a Conic

Determining whether a given point lies inside, on, or outside a conic section such as an ellipse involves substituting the point's coordinates into the standard equation or inequality of the conic. The resulting condition or inequality provides insight into the spatial relationship between the point and the conic section. This concept is important for understanding geometric configurations and for solving problems where constraints are imposed by the location of specific points relative to the conic.

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