91Ó°ÊÓ

Understanding circle equations is crucial in coordinate geometry. The general equation of a circle is given by \[(x-h)^2 + (y-k)^2 = r^2\] where \(h\) and \(k\) are the coordinates of the center, and \(r\) is the radius. By expanding the terms, this equation can be expressed in a different form: \[x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0\]. This is what we call the general form of a circle equation.In our problem, the circles have a radius of 5 units and touch at the point \((1, 2)\). This information is used to determine the specific equations of the circles. Since both circles share a tangent, we're specifically working on finding out how their individual equations come to be derived from the shared tangent and point of tangency.

A tangent to a circle is a line that touches the circle at exactly one point. The equation of the tangent can be derived from the circle's equation by using the normal form, \[ax + by = ±r\], where \(a\) and \(b\) are coefficients from the tangent's general line equation and \(r\) is the radius.In our problem, the common tangent to the two circles is \(4x + 3y = 10\). Hence, this line serves as a unique tangent to both circles, touching them at the point \((1, 2)\). Since the line is a tangent, the centers of the circles must lie on a line that is perpendicular to this tangent at the point of contact. This perpendicularity plays a critical role in ascertaining the precise location of the centers of the circles and subsequently their respective equations.

Geometric loci are sets of points that satisfy particular conditions. In the context of circles, the locus can often refer to the set of points that make up the circle itself or related features.In the problem at hand, we consider the locus of points that lie on the perpendicular line to the common tangent at the point \((1, 2)\). The line's equation is derived as \(3x + 4y = 11\). All potential center points for the two circles must fall on this line since it forms the natural geometric locus for their centers based on the problem's conditions.This locus is a crucial link for determining where exactly the centers of the circles need to be situated in relation to the tangent and the defined point of contact.

In coordinate geometry, intersection points between circles can vary. A single point of intersection, often referred to as "touching", indicates that the circles are tangent to each other, either externally or internally.For our circles that intersect at \((1, 2)\), each is tangent to the other. This unique point of intersection allows them to share a common tangent line, establishing the relationship between their respective equations. The relationship is reflected in their centers being exactly the sum of their radii apart, which for this case reinforces that the circles touch externally with each having a radius of 5 units.Analyzing how and where circles intersect aids in visualizing their potential tangents and how these tangents influence their equations. This becomes the essence of solving the original exercise, ensuring the identified equations reflect this association accurately.