The relationship between circles and straight lines is an important topic in analytic geometry, which not only involves basic geometric concepts, but also integrates the knowledge of algebra and analytic geometry, and is a bridge between middle school mathematics and advanced mathematics. Today, we will analyze the position relationship between a circle and a straight line through a problem, and explain the solution process in detail.
Problem description: Given a circle (c) the equation is ((x-2) 2(y3) 2=25) and a straight line (l) is (3x4yk=0). Solve what value (k) takes, the line (l) is tangent to the circle (c).
Steps: Step 1: Understand the problem.
We need to understand the basic geometric elements of the question. The center of the circle (c) is ((2,-3)) and the radius (r=5). The slope of the straight line (l) is (-frac) and the intercept (k) is the unknown we need to solve. The condition for a circle to be tangent to a straight line is that the distance from the line to the center of the circle is equal to the radius of the circle.
Step 2: Distance formula from straight line to center of circle.
The formula for the distance from a straight line to a point is (d=frac}), where (a), (b), and (c) are the coefficients in the equation of a straight line (axbyc=0), and ((x1,y1)) are the coordinates of the point. In this problem, the coordinates of the center of the line (l) are (a=3), (b=4), and (c=-k), which is ((2,-3)).
Step 3: Substitute the formula to solve.
Substituting the coordinates of the line (l) and the center of the circle into the formula for the distance from the line to the point, we get:
d=frac}=frac=frac。
Since the circle is tangent to the straight line, this distance (d) must be equal to the radius of the circle (r=5). We can get the equation:.
frac=5
Step 4: Solve the equation to find the value of (k).
Solving this value equation gives us two possible cases:
6k = 25 or -6k = -25
Solving these two equations gives us the value of two (k):
k1 = -31 and k2 = 19
When (k=-31) or (k=19), the line (l) is tangent to the circle (c). The solution to this problem not only shows the basic method of determining the relationship between the position of circles and straight lines, but also integrates the knowledge of geometry and algebra, showing the beauty and practicality of mathematics.
Through the analysis of this problem, we can see how the position relationship between circles and lines in analytic geometry is determined by algebraic methods, and also emphasize the importance of basic geometric concepts and algebraic skills. Hopefully, this analysis will help you better understand and master the relevant mathematical knowledge.