For the circles x^2+y^2+6 x-16=0 and x^2+y^2-2 x+6 y-6=0 (a)...

For the circles $x^{2}+y^{2}+6 x-16=0$ and $x^{2}+y^{2}-2 x+6 y-6=0$
(a) Number of common tangents is 2
(b) Number of common tangents is 3
(c) Distance of the point $(1,2)$ from the radical axis of the circles is $\frac{7}{5}$
(d) the circles touch internally.

Key Concepts

Standard Form of a Circle

The standard form of a circle’s equation is essential for identifying its center and radius, which are pivotal for many geometric analyses. Transforming a circle’s equation from the general form into the standard form by completing the square allows for easier interpretation and further computations, such as determining the positional relationships between circles or calculating distances relevant to the problem.

Radical Axis

The radical axis of two circles is the line containing all points that have equal power with respect to both circles. It plays a crucial role in problems involving relationships between circles, such as finding the common tangents or determining distances from specific points to this line, and it provides insight into the intersecting properties of the circles.

Common Tangents

Common tangents are the straight lines that touch two circles without crossing them except at the point of tangency. Understanding the concept of tangency and the conditions for the existence of common tangents, including the number of tangents possible given the circles' positions and sizes, is fundamental in solving geometric problems related to circle configurations.

Distance from a Point to a Line

Calculating the distance from a point to a line is a key geometric operation, often used to determine the closest distance between geometric objects or verify specific distance criteria. This concept is employed when examining the relationship of a point with a line derived from other entities, such as the radical axis, which further informs the positioning of other geometric constructs.

Transcript

Please give Ace some feedback