设r > 0, 则两圆(x - 1)^2 + (y + 3)^2 = r^2 与 x^2 + y^2 = 16的位置关系不可能是()A. 相切 B. 相交 C. 内切和内含 D. 外切和外离 ,生成这个题的讲解视频---**Extracted Content:**
**Question 1:**
**Question Stem:**
设r > 0, 则两圆(x - 1)^2 + (y + 3)^2 = r^2 与 x^2 + y^2 = 16的位置关系不可能是()
**Mathematical Formulas:**
Circle 1: (x - 1)^2 + (y + 3)^2 = r^2
Circle 2: x^2 + y^2 = 16
**Options:**
A. 相切 (Tangent)
B. 相交 (Intersecting)
C. 内切和内含 (Internally tangent and internally contained)
D. 外切和外离 (Externally tangent and externally disjoint)
视频信息
答案文本
视频字幕
We have two circles to analyze. Circle 1 has center at (1, -3) with radius r, and Circle 2 has center at origin with radius 4. The distance between centers is square root of 10, approximately 3.16. This distance will be crucial for determining possible position relationships.
今天我们来解决一道关于两圆位置关系的选择题。题目给出两个圆的方程，要求判断在r大于0的条件下，这两个圆的位置关系中哪一种是不可能的。
首先分析两个圆的基本信息。圆1的圆心在(1, -3)，半径为r。圆2的圆心在原点(0, 0)，半径为4。两圆心之间的距离d等于根号10，约等于3.16。
两圆的位置关系完全由圆心距d与两圆半径的大小关系决定。当d大于两半径之和时，两圆外离；当d等于两半径之和时，两圆外切；当d在两半径差的绝对值与和之间时，两圆相交；当d等于两半径差的绝对值时，两圆内切；当d小于两半径差的绝对值时，一圆在另一圆内部。
现在分析各种位置关系的条件。由于d等于根号10约为3.16，小于4，因此外离是不可能的，因为这需要r小于负数。外切需要r等于4加根号10。相交需要r在一定范围内。内切有两种情况。内含也需要满足特定条件。
通过分析可以发现，选项A、B、C都是可能的。但是选项D中的外离是不可能的，因为外离需要r小于根号10减4，这是一个负数，与题目条件r大于0矛盾。因此答案是D。
Now we apply the theoretical framework to our specific problem. With distance d equals square root of 10 and radius 4 for the second circle, we can analyze each relationship. External separation requires r less than square root of 10 minus 4, which is negative, making it impossible since r must be positive. However, external tangency, intersection, internal tangency, and internal containment are all possible for appropriate values of r.
Let's visualize how the circle positions change as radius r varies. The blue circle is fixed at the origin with radius 4, while the red circle has center at (1, -3) with variable radius r. We can see internal containment for small r, internal tangency at r equals 4 minus square root of 10, intersection for intermediate values, and external tangency at r equals 4 plus square root of 10. External separation would require negative r, which is impossible.
Now let's systematically evaluate each option. Option A, tangent relationships, is possible with both internal and external tangency at specific r values. Option B, intersecting circles, is possible for r values in the appropriate range. Option C, internal tangent and internal containment, is also possible. However, Option D contains external separation, which requires r less than square root of 10 minus 4, giving a negative value. This contradicts the given condition that r is positive. Therefore, the answer is D.