If the circles `x^2+y^2-16x-20y+164=r^2` and `(x-4)^2+(y-7)^2=36` intersect at two points then (a) `1ltrlt11` (b) `r=11` (c) `rgt11` (d) `0ltrlt1`

Options

A. `0 lt r lt 1`
B. `1 lt r lt 11`
C. `r lt 11`
D. `r=11`

Answer

To solve the problem, we need to analyze the two circles given by their equations and determine the conditions under which they intersect at two points.

### Step 1: Rewrite the first circle equation
The first circle is given by:
\[ x^2 + y^2 - 16x - 20y + 164 = r^2 \]

We can rewrite this in standard form by completing the square for both \(x\) and \(y\).

1. For \(x\):
\[ x^2 - 16x = (x - 8)^2 - 64 \]

2. For \(y\):
\[ y^2 - 20y = (y - 10)^2 - 100 \]

Now substituting these back into the equation:
\[ (x - 8)^2 - 64 + (y - 10)^2 - 100 + 164 = r^2 \]
This simplifies to:
\[ (x - 8)^2 + (y - 10)^2 = r^2 + 64 + 100 - 164 \]
\[ (x - 8)^2 + (y - 10)^2 = r^2 \]

Thus, the center of the first circle is \(A(8, 10)\) and its radius is \(r\).

### Step 2: Analyze the second circle
The second circle is given by:
\[ (x - 4)^2 + (y - 7)^2 = 36 \]
This indicates that the center of the second circle is \(B(4, 7)\) and its radius is \(6\) (since \(36 = 6^2\)).

### Step 3: Calculate the distance between the centers
Now, we need to find the distance \(AB\) between the centers \(A(8, 10)\) and \(B(4, 7)\):
\[ AB = \sqrt{(8 - 4)^2 + (10 - 7)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 4: Apply the intersection condition
For the circles to intersect at two points, the following condition must hold:
\[ |r_1 - r_2| < AB < r_1 + r_2 \]
Where \(r_1 = r\) (radius of the first circle) and \(r_2 = 6\) (radius of the second circle).

Substituting the values:
1. The left side of the inequality:
\[ |r - 6| < 5 \]
This leads to two inequalities:
\[ -5 < r - 6 < 5 \]
Which simplifies to:
\[ 1 < r < 11 \]

2. The right side of the inequality:
\[ 5 < r + 6 \]
This simplifies to:
\[ r > -1 \quad \text{(which is always true since } r \text{ is a radius)} \]

### Conclusion
From the inequalities derived, we conclude:
\[ 1 < r < 11 \]

Thus, the correct answer is option (a) \(1 < r < 11\).