The angle between the radical axis and the line joining the...

The angle between the radical axis and the line joining the centres of the circles $x^{2}+y^{2}+2 g x+2 f y+c=0$ and $x^{2}+y^{2}$ $+2 g_{1} x+2 f_{1} y+c_{1}=0$ is
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{2}$
(c) 0
(d) $\frac{\pi}{6}$

Key Concepts

Circle Equation

A circle can be represented in general form by an equation such as x² + y² + 2gx + 2fy + c = 0. In this form, the center of the circle is given by the point (-g, -f) and the radius is determined by the expression ?(g² + f² - c), provided that the value under the square root is positive. This representation is fundamental in coordinate geometry for analyzing and solving problems involving circles.

Radical Axis

The radical axis of two circles is the set of points that have equal power with respect to both circles. It is a straight line and possesses significant geometric properties, such as its consistent perpendicularity to the line joining the centers of the two circles. This concept is crucial for understanding the relative positions and intersections of circles.

Perpendicularity of Radical Axis and Line Joining Centers

A key property in circle geometry is that the radical axis of two circles is perpendicular to the line joining their centers. This perpendicularity arises from the symmetry in the definition of the radical axis and is widely used in proofs and problem-solving to relate circle positions and their intersections.

Angle Between Two Lines

The concept of the angle between two lines involves determining the measure of the intersection between them. When slopes are available, the angle can be computed using the formula tan? = |(m? - m?)/(1 + m?m?)|. This concept is essential in coordinate geometry and provides the mathematical basis for understanding the orientation of lines in the plane.

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