The minimum value of `{(r+5 -4|cos theta|)^(2) +(r-3|sin theta|)^(2)} AA r, theta in R` is

Options

A. 0
B. 2
C. 3
D. None of these

Explanation

To find the minimum value of the expression \( z = (r + 5 - 4 |\cos \theta|)^2 + (r - 3 |\sin \theta|)^2 \), we will follow these steps:

### Step 1: Rewrite the expression
Let \( z = (r + 5 - 4 |\cos \theta|)^2 + (r - 3 |\sin \theta|)^2 \).

### Step 2: Introduce new variables
Let \( x = r + 5 \) and \( y = r \). Then, we can express \( r \) in terms of \( x \) as \( r = x - 5 \). Thus, we can rewrite \( z \) in terms of \( x \):
\[ z = (x - 4 |\cos \theta|)^2 + ((x - 5) - 3 |\sin \theta|)^2 \]

### Step 3: Expand the expression
Now, we expand the expression:
\[ z = (x - 4 |\cos \theta|)^2 + (x - 5 - 3 |\sin \theta|)^2 \]
Expanding both squares:
\[ = (x^2 - 8x |\cos \theta| + 16 \cos^2 \theta) + (x^2 - 10x + 25 - 6x |\sin \theta| + 9 \sin^2 \theta) \]
Combining like terms gives:
\[ z = 2x^2 - (8 |\cos \theta| + 10 + 6 |\sin \theta|) x + (16 \cos^2 \theta + 25 + 9 \sin^2 \theta) \]

### Step 4: Find the minimum value
To find the minimum value of \( z \), we can treat it as a quadratic function in \( x \):
\[ z = 2x^2 - (8 |\cos \theta| + 10 + 6 |\sin \theta|) x + (16 \cos^2 \theta + 25 + 9 \sin^2 \theta) \]
The minimum value of a quadratic \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). Here, \( a = 2 \) and \( b = -(8 |\cos \theta| + 10 + 6 |\sin \theta|) \):
\[ x_{\text{min}} = \frac{8 |\cos \theta| + 10 + 6 |\sin \theta|}{4} \]

### Step 5: Substitute back to find \( z \)
Substituting \( x_{\text{min}} \) back into the expression for \( z \) will give us the minimum value. However, we can also analyze the terms:
1. The minimum of \( |\cos \theta| \) is 0 and the maximum is 1.
2. The minimum of \( |\sin \theta| \) is 0 and the maximum is 1.

Thus, we can evaluate \( z \) at critical points:
- When \( |\cos \theta| = 1 \) and \( |\sin \theta| = 0 \):
\[ z = (r + 5 - 4)^2 + (r - 0)^2 = (r + 1)^2 + r^2 \]
- When \( |\cos \theta| = 0 \) and \( |\sin \theta| = 1 \):
\[ z = (r + 5)^2 + (r - 3)^2 \]

### Step 6: Analyze the results
By analyzing the values for different combinations of \( |\cos \theta| \) and \( |\sin \theta| \), we can find that the minimum value occurs when the two curves are tangent to each other.

### Conclusion
After evaluating the expression and considering the geometry of the situation, we find that the minimum value of \( z \) is \( 0 \).