If the tangent at a point θon an ellipse meets the auxiliary circle of the ellipse at two points

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If the tangent at a point θon an ellipse meets the auxiliary...

If the tangent at a point $\theta$ on an ellipse meets the auxiliary circle of the ellipse at two points, which subtend a right angle at the centre, the eccentricity of the ellipse is (a) $1+\sin ^{2} \theta$ (b) $\sin ^{2} \theta$ (c) $\left(1+\sin ^{2} \theta\right)^{-1 / 2}$ (d) $(1+\sin \theta)^{1 / 2}$

If the tangent at a point $\theta$ on an ellipse meets the auxiliary circle of the ellipse at two points, which subtend a right angle at the centre, the eccentricity of the ellipse is
(a) $1+\sin ^{2} \theta$
(b) $\sin ^{2} \theta$
(c) $\left(1+\sin ^{2} \theta\right)^{-1 / 2}$
(d) $(1+\sin \theta)^{1 / 2}$

Key Concepts

Ellipse

An ellipse is a conic section defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. It is characterized by its semi-major axis and semi-minor axis, and it exhibits properties that differ significantly from a circle, particularly when dealing with tangents and auxiliary constructions.

Eccentricity

Eccentricity is a number between 0 and 1 that measures how much an ellipse deviates from being a circle. In an ellipse, it is defined as the ratio of the distance from the center to a focus over the semi-major axis, and it plays a crucial role in determining the shape and geometric properties of the ellipse.

Auxiliary Circle

The auxiliary circle of an ellipse is a circle drawn with the same center as the ellipse and with a radius equal to the semi-major axis of the ellipse. This concept is used to relate the ellipse to a circle, simplifying some aspects of its geometry and is particularly useful when analyzing properties involving tangents and angles.

Tangent Line to an Ellipse

A tangent line to an ellipse is a straight line that touches the ellipse at exactly one point. The properties of tangents are essential in problems where lines intersect other conic sections, and in this context, it is used to determine relationships between points on the ellipse, its auxiliary circle, and angles at the center.

Angle Subtension in a Circle

In circle geometry, the angle subtended at the center by a chord is directly connected to the chord's length and the circle's radius. When two points on a circle subtend a right angle at the center, they satisfy a specific orthogonality condition, which can be exploited using vector dot products or trigonometric identities to relate different parameters, such as the position of a tangent point on an ellipse and its eccentricity.

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