The number of common tangents to the circles x^2+y^2 -4 x-6 y-12=0 and x^2+y^2+6 x+18 y+26=0, is

step 1 Then we are given two circle equations. C1 is given as x square plus y square minus 4x minus 6y

The number of common tangents to the circles x^2+y^2 -4 x-6...

Key Concepts

Standard Form of a Circle

This concept involves rewriting the general quadratic equation of a circle into its standard form by completing the square. The standard form clearly reveals the circle's center and radius, which are critical for understanding its geometric properties and relationships with other figures.

Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. Understanding the properties of tangents, such as the perpendicularity between the radius at the point of contact and the tangent line, is essential in solving problems involving lines that interact with circles.

Common Tangents

Common tangents are lines that are tangent to two circles simultaneously. Identifying and counting common tangents involves analyzing the relative positions of the circles, such as whether they are separate, intersecting, or one contained within the other, since these configurations determine the number of possible common tangents.

Distance between Circle Centers

The distance between the centers of two circles is a key measurement that helps classify the positional relationships between the circles. It is used in conjunction with the radii to determine if the circles are separate, touching, or overlapping, which further influences the number of common tangents possible.

Relative Positioning of Circles

This concept refers to the spatial configuration of two circles based on the distance between their centers and their radii. Analyzing whether circles are externally tangent, internally tangent, intersecting, or disjoint allows one to determine the number and nature of tangents and other geometric relationships between the circles.

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