Let A,B, and C be three sets such that
`A={(x,y)|(x)/(cos theta)=(y)/(sintheta)=5,"where" 'theta'"is parameter"}`
`B= {(x,y)|(x-3)/(cos phi)=(y-4)/(sin phi)=r}`
`C= { (x,y)|(x-3)^(2)+(y-4)^(2)leR^(2)}`
If `A capC =A`, then minimum value of R is
(a)5
(b)6
(c)10
(d)11

Options

A. 5
B. 6
C. 10
D. 11

Explanation

To solve the problem, we need to analyze the sets A, B, and C given in the question.

### Step 1: Analyze Set A
Set A is defined as:
\[ A = \{(x,y) | \frac{x}{\cos \theta} = \frac{y}{\sin \theta} = 5\} \]

From this, we can derive the parametric equations:
\[ x = 5 \cos \theta, \quad y = 5 \sin \theta \]

This represents a circle with center at the origin (0,0) and radius 5. The equation of this circle is:
\[ x^2 + y^2 = 5^2 \quad \Rightarrow \quad x^2 + y^2 = 25 \]

### Step 2: Analyze Set B
Set B is defined as:
\[ B = \{(x,y) | \frac{x-3}{\cos \phi} = \frac{y-4}{\sin \phi} = r\} \]

From this, we can derive the parametric equations:
\[ x - 3 = r \cos \phi, \quad y - 4 = r \sin \phi \]

This represents a circle with center at (3,4) and radius r. The equation of this circle is:
\[ (x - 3)^2 + (y - 4)^2 = r^2 \]

### Step 3: Analyze Set C
Set C is defined as:
\[ C = \{(x,y) | (x-3)^2 + (y-4)^2 \leq R^2\} \]

This is the area inside and on the circle centered at (3,4) with radius R.

### Step 4: Condition Given in the Problem
We are given that \( A \cap C = A \). This means that every point in set A must also be in set C. For this to happen, the circle defined by set A must touch or be inside the circle defined by set C.

### Step 5: Determine the Distance Between the Centers
The center of circle A is at (0,0) and the center of circle B (and C) is at (3,4). We need to find the distance \( d \) between these two centers:
\[ d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Condition for Internal Tangency
For the circles to touch internally, the following condition must hold:
\[ R - 5 = d \]
Substituting the value of \( d \):
\[ R - 5 = 5 \quad \Rightarrow \quad R = 10 \]

### Conclusion
Thus, the minimum value of \( R \) such that \( A \cap C = A \) is:
\[ \boxed{10} \]