Locus of point of intersection of perpendicular tangents to...

Locus of point of intersection of perpendicular tangents to the circle $x^{2}+y^{2}-4 x-6 y-12=0$
(a) $x^{2}+y^{2}-4 x-6 y+37=0$
(b) $x^{2}+y^{2}-4 x-6 y-37=0$
(c) $x^{2}+y^{2}+4 x+6 y-37=0$
(d) $x^{2}+y^{2}-4 x-6 y-25=0$

Key Concepts

Circle

A circle is defined as the set of all points in a plane that are at a fixed distance (the radius) from a given point (the center). The standard form of the circle's equation can be derived by completing the square when starting with a general quadratic equation, helping to identify key features such as the center and the radius.

Tangents to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. The condition for a line to be tangent to a circle is found by ensuring that the distance from the center of the circle to the line equals the radius, typically resulting in a discriminant of zero in the corresponding quadratic equation when the line’s equation is substituted into the circle’s equation.

Perpendicular Tangents

Perpendicular tangents refer to two tangent lines to a circle that intersect at a right angle. The geometric condition for two lines to be perpendicular is that the product of their slopes is -1. This additional relation helps in determining the specific positioning of tangents relative to the circle.

Director Circle

The director circle of a given circle is the locus of the points of intersection of pairs of perpendicular tangents drawn to the circle. For any circle, the director circle is a significant concept in geometry because it provides an elegant description of the relationship between the original circle and the family of its tangents, often leading to a new circle whose equation can be deduced through geometric conditions.

Locus

In geometry, a locus is a set of points that satisfy a particular condition or a series of conditions. Determining the locus involves finding a general equation that represents all these points, which in this context is done by examining the intersection points of perpendicular tangents to the original circle.

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