(i) From any point on the circle x^2+y^2+2 g x+2 f y+c=0,...

(i) From any point on the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$, tangents are drawn to the circle $x^{2}+y^{2}+2 g x+2 f y+c \sin ^{2} \alpha$ $+\left(\mathrm{g}^{2}+\mathrm{f}^{2}\right) \cos ^{2} \alpha=0$ Find the angle between these tangents. (ii) If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is the director circle of the circle $x^{2}+y^{2}+2 g x+2 f y+k=0$, find the value of $\mathrm{k}$.

(i) From any point on the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$, tangents are drawn to the circle $x^{2}+y^{2}+2 g x+2 f y+c \sin ^{2} \alpha$ $+\left(\mathrm{g}^{2}+\mathrm{f}^{2}\right) \cos ^{2} \alpha=0$ Find the angle between these tangents.
(ii) If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is the director circle of the circle $x^{2}+y^{2}+2 g x+2 f y+k=0$, find the value of $\mathrm{k}$.

Key Concepts

Circle Equation

A circle in the Cartesian plane is typically expressed in the form x² + y² + 2gx + 2fy + c = 0. This form, which can be manipulated by completing the square, reveals the circle’s center (–g, –f) and its radius (?(g² + f² – c)). Understanding the standard equation of a circle is fundamental in analyzing its geometric properties and relationships with other curves or lines.

Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point and is perpendicular to the radius at the point of contact. The concept of a tangent is pivotal when studying circle geometry, as tangents have unique properties that lead to important conditions and formulas, such as those determining the length of tangents or the angles formed between them when drawn from the same external point.

Angle Between Tangents

When two tangents are drawn from a single external point to a circle, the angle between these tangents can be determined through geometric or trigonometric relationships. One common approach involves relating the angle to the circle's radius and the distance from the external point to the circle's center. This concept is crucial in various problems where the measure of that angle is connected to other parameters of the circle.

Director Circle

The director circle of a given circle is defined as the locus of points from which two tangents drawn to the given circle are perpendicular to each other. For a circle with center (–g, –f) and radius r, its director circle is often given by a related circle equation and is used to explore orthogonality conditions and special relationships between circles. This concept is essential when establishing the link between a circle and its tangents, particularly in problems that involve right-angle properties.

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