If A((3)/(√(2)), √(2)), B(-(3)/(√(2)), √(2)), C(-(3)/(√(2)),-√(2)) and D(3 cosθ, 2 sinθ) are fou

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If A((3)/(√(2)), √(2)), B(-(3)/(√(2)), √(2)),...

If $\mathrm{A}\left(\frac{3}{\sqrt{2}}, \sqrt{2}\right), \mathrm{B}\left(-\frac{3}{\sqrt{2}}, \sqrt{2}\right), \mathrm{C}\left(-\frac{3}{\sqrt{2}},-\sqrt{2}\right)$ and $\mathrm{D}(3 \cos \theta, 2 \sin \theta)$ are four points, then the value of $\theta$ for which the area of quadrilateral $\mathrm{ABCD}$ is maximum, $\left(\frac{3 \pi}{2} \leq 0 \leq 2 \pi\right)$ is (A) $2 \pi-\sin ^{-1} \frac{1}{3}$ (B) $\frac{7 \pi}{4}$ (C) $2 \pi-\cos ^{-1} \frac{3}{\sqrt{85}}$ (D) None of these

If $\mathrm{A}\left(\frac{3}{\sqrt{2}}, \sqrt{2}\right), \mathrm{B}\left(-\frac{3}{\sqrt{2}}, \sqrt{2}\right), \mathrm{C}\left(-\frac{3}{\sqrt{2}},-\sqrt{2}\right)$ and $\mathrm{D}(3 \cos \theta, 2 \sin \theta)$ are four points, then the value of $\theta$ for which the area of quadrilateral $\mathrm{ABCD}$ is maximum, $\left(\frac{3 \pi}{2} \leq 0 \leq 2 \pi\right)$ is
(A) $2 \pi-\sin ^{-1} \frac{1}{3}$
(B) $\frac{7 \pi}{4}$
(C) $2 \pi-\cos ^{-1} \frac{3}{\sqrt{85}}$
(D) None of these

Key Concepts

Optimization Techniques in Calculus

Optimization in calculus involves finding the maximum or minimum values of a function, often by setting its derivative to zero and analyzing critical points. This concept is vital for problems where one must determine the parameter value that maximizes or minimizes a specific quantity, such as the area of a geometrically defined shape.

Trigonometric Functions and Identities

Trigonometric functions, such as sine and cosine, and their accompanying identities are crucial for expressing coordinates in problems involving circular or elliptical motion. Mastery of these functions allows one to simplify and manipulate expressions, which is especially important when optimizing quantities that depend on an angle.

Coordinate Geometry

Coordinate geometry involves representing geometric figures using a coordinate system, typically the Cartesian plane. This approach enables one to perform algebraic calculations to determine properties such as distances, slopes, and areas, and is essential when dealing with points defined by coordinates.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted by t or ?. This concept is particularly useful for describing curves like circles and ellipses, where each point's coordinates are generated from trigonometric functions, allowing for easier analysis of how a point moves as the parameter varies.

Area Calculation Using the Shoelace Formula

The shoelace formula provides a systematic method to calculate the area of a polygon when the coordinates of its vertices are known. This technique is fundamental in coordinate geometry as it transforms a geometric property (area) into an algebraic expression, facilitating further operations such as optimization.

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