The circles `x ^(2) + y ^(2) -10 x + 16=0 and x ^(2) + y ^(2) =r ^(2)` intersect each other in two distinct points if

Options

A. `r lt 2`
B. `r gt 8 `
C. `2 lt r lt 8`
D. `2 ge r ge 8`

Answer

To determine the conditions under which the circles \( x^2 + y^2 - 10x + 16 = 0 \) and \( x^2 + y^2 = r^2 \) intersect at two distinct points, we will follow these steps:

### Step 1: Rewrite the first circle's equation
The first circle's equation is given as:
\[ x^2 + y^2 - 10x + 16 = 0 \]
We can rearrange this to identify the center and radius. Completing the square for the \(x\) terms:
\[ (x^2 - 10x) + y^2 + 16 = 0 \]
\[ (x - 5)^2 - 25 + y^2 + 16 = 0 \]
\[ (x - 5)^2 + y^2 = 9 \]
This shows that the center of the first circle \(C_1\) is at \((5, 0)\) and the radius \(r_1\) is:
\[ r_1 = \sqrt{9} = 3 \]

### Step 2: Identify the second circle's properties
The second circle is given by:
\[ x^2 + y^2 = r^2 \]
This circle is centered at the origin \((0, 0)\) with radius \(r\).

### Step 3: Determine the distance between the centers
The distance \(d\) between the centers of the two circles \(C_1(5, 0)\) and \(C_2(0, 0)\) is calculated as:
\[ d = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25} = 5 \]

### Step 4: Apply the condition for intersection
For two circles to intersect at two distinct points, the following condition must hold:
\[ |r_1 - r| < d < r_1 + r \]
Substituting the known values:
1. \(r_1 = 3\)
2. \(d = 5\)

This gives us two inequalities:
1. \( |3 - r| < 5 \)
2. \( 5 < 3 + r \)

### Step 5: Solve the inequalities
**From the first inequality:**
\[ -5 < 3 - r < 5 \]
This can be split into two parts:
1. \(3 - r > -5 \Rightarrow r < 8\)
2. \(3 - r < 5 \Rightarrow r > -2\) (which is always true since \(r\) is a radius)

**From the second inequality:**
\[ 5 < 3 + r \Rightarrow r > 2 \]

### Step 6: Combine the results
Combining the results from both inequalities:
\[ 2 < r < 8 \]

### Conclusion
The circles \(x^2 + y^2 - 10x + 16 = 0\) and \(x^2 + y^2 = r^2\) intersect at two distinct points if:
\[ 2 < r < 8 \]