If θis the angle subtended at P(2,3) by the circle x^2+y^2+2 x-3 y+1=0, then (a) cot((θ)/(2))=2

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If θis the angle subtended at P(2,3) by the circle x^2+y^2+2...

If $\theta$ is the angle subtended at $\mathrm{P}(2,3)$ by the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{x}-3 \mathrm{y}+1=0$, then (a) $\cot \left(\frac{\theta}{2}\right)=2$ (b) $\tan \left(\frac{\theta}{2}\right)=\frac{1}{\sqrt{2}}$ (c) $\tan \theta=\frac{4}{3}$ (d) $\sec \theta=3$

If $\theta$ is the angle subtended at $\mathrm{P}(2,3)$ by the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{x}-3 \mathrm{y}+1=0$, then
(a) $\cot \left(\frac{\theta}{2}\right)=2$
(b) $\tan \left(\frac{\theta}{2}\right)=\frac{1}{\sqrt{2}}$
(c) $\tan \theta=\frac{4}{3}$
(d) $\sec \theta=3$

Key Concepts

Trigonometric Half-Angle Identities

Trigonometric half-angle identities allow one to express sine, cosine, tangent, and cotangent of half of an angle in terms of the original angle or in relation to the geometry of a circle. They are particularly useful in problems where the angle subtended by a circle is divided into two equal parts, enabling the formulation of relationships between the circle’s parameters and the angle measurements.

Angle Subtended by a Circle at an External Point

This concept pertains to the angle formed at an external point by the two tangents drawn to the circle. The measure of this angle is directly related to the circle’s radius and the distance from the external point to the center. Using geometrical and trigonometric relationships, one can derive expressions for the half-angle, which are essential for establishing tangent or cotangent values as seen in various problems.

Distance in Euclidean Geometry

The distance formula is necessary to compute the distance between two points in the plane. In problems involving circles and external points, calculating the distance from the point to the circle’s center helps determine whether the point lies inside, on, or outside the circle, which in turn influences the nature of the angles and tangents drawn.

Circle Equation and Standard Form

This concept involves rewriting a circle's general equation into its standard form to identify the center and radius. By completing the square for the x and y terms, one can determine the circle's geometric parameters, which are crucial for solving problems related to distances, tangents, and angles associated with the circle.

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