The tangent at P(5 cosθ, 3 sinθ) on the ellipse (x^2)/(25)+(y^2)/(9)=1 meets the auxiliary circl

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The tangent at P(5 cosθ, 3 sinθ) on the ellipse...

The tangent at $\mathrm{P}(5 \cos \theta, 3 \sin \theta)$ on the ellipse $\frac{\mathrm{x}^{2}}{25}+\frac{\mathrm{y}^{2}}{9}=1$ meets the auxiliary circle $\mathrm{x}^{2}+\mathrm{y}^{2}=25$ at two points $\mathrm{A}$ and $\mathrm{B}$. If AB subtends $90^{\circ}$ at the centre of the ellipse, then (a) $\sin ^{2} \theta=\frac{9}{16}$ (b) $\sin ^{2} \theta=\frac{16}{25}$ (c) $\mathrm{e}^{2}=\left(1+\sin ^{2} \theta\right)^{-1}$ (d) $\mathrm{e}^{2}=\left(1+\cos ^{2} \theta\right)^{-1}$

The tangent at $\mathrm{P}(5 \cos \theta, 3 \sin \theta)$ on the ellipse $\frac{\mathrm{x}^{2}}{25}+\frac{\mathrm{y}^{2}}{9}=1$ meets the auxiliary circle $\mathrm{x}^{2}+\mathrm{y}^{2}=25$ at two points $\mathrm{A}$ and $\mathrm{B}$.
If AB subtends $90^{\circ}$ at the centre of the ellipse, then
(a) $\sin ^{2} \theta=\frac{9}{16}$
(b) $\sin ^{2} \theta=\frac{16}{25}$
(c) $\mathrm{e}^{2}=\left(1+\sin ^{2} \theta\right)^{-1}$
(d) $\mathrm{e}^{2}=\left(1+\cos ^{2} \theta\right)^{-1}$

Key Concepts

Ellipse Geometry

An ellipse is a conic section defined by the set of points for which the sum of the distances from two fixed points (the foci) is constant. Its standard form, written as x²/a² + y²/b² = 1, specifies the lengths of its semi-major and semi-minor axes, and understanding its geometric properties is essential in many analytical problems.

Auxiliary Circle

The auxiliary circle of an ellipse is the circle that shares the same center as the ellipse and has a radius equal to the length of the semi?major axis of the ellipse. It provides a geometric tool to relate angular parameters with the coordinates on the ellipse and is often used to simplify problems involving tangency and intersections.

Tangent to a Conic

A tangent to a conic section, such as an ellipse, is a line that touches the curve at exactly one point. The tangent line at a point on the ellipse, especially when expressed in its parametric form, can be used to determine intersections with other curves and to derive important geometric relationships.

Chord Subtending an Angle at the Centre

When a chord of a circle subtends a specific angle at the centre, it creates a defined geometrical relation between the chord’s endpoints. For example, if the chord subtends a 90° angle, the line segments from the centre to these endpoints are orthogonal. This concept is applied to connect the geometry of tangents intersecting the circle with angular relationships.

Eccentricity of an Ellipse

The eccentricity of an ellipse, given by e = ?(1 - (b²/a²)) for an ellipse in standard form, indicates how much the ellipse deviates from being a circle. It is a fundamental parameter in the study of conic sections and often appears in relationships that tie together other geometric properties, such as angles and tangency conditions.

Parametric Representation of an Ellipse

The parametric form of an ellipse, typically expressed as (a cos ?, b sin ?), offers a convenient way to represent points on the ellipse. This representation simplifies the derivation of the tangent at a given point and aids in solving problems that involve intersections and angular subtensions involving the ellipse and related figures.

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