The number of points with integral coordinates that lie in the interior of the region common to the circle `x^(2)+y^(2)=16` and the parabola `y^(2)=4x`, is

Options

A. 8
B. 10
C. 16
D. none of these

Explanation

To find the number of points with integral coordinates that lie in the interior of the region common to the circle \( x^2 + y^2 = 16 \) and the parabola \( y^2 = 4x \), we can follow these steps:

### Step 1: Understand the equations
The equation of the circle \( x^2 + y^2 = 16 \) represents a circle centered at the origin (0,0) with a radius of 4. The equation of the parabola \( y^2 = 4x \) opens to the right.

### Step 2: Identify the region of interest
We need to find the points that lie inside both the circle and the parabola. For a point \( (x, y) \) to be inside the circle, it must satisfy:
\[ x^2 + y^2 < 16 \]
For the parabola, it must satisfy:
\[ y^2 < 4x \]

### Step 3: Determine the range for \( x \)
Since the parabola opens to the right, \( x \) must be non-negative. Additionally, since we are looking for points inside the circle, \( x \) must also be less than 4 (the x-coordinate of the circle's edge). Therefore:
\[ 0 < x < 4 \]
The integral values of \( x \) in this range are \( 1, 2, 3 \).

### Step 4: Find corresponding \( y \) values for each \( x \)
Now we will find the possible integral values of \( y \) for each integral value of \( x \).

#### For \( x = 1 \):
1. From the circle:
\[ y^2 < 16 - 1^2 \Rightarrow y^2 < 15 \Rightarrow -\sqrt{15} < y < \sqrt{15} \]
The integral values of \( y \) are \( -3, -2, -1, 0, 1, 2, 3 \) (7 values).

2. From the parabola:
\[ y^2 < 4 \cdot 1 \Rightarrow y^2 < 4 \Rightarrow -2 < y < 2 \]
The integral values of \( y \) are \( -1, 0, 1 \) (3 values).

3. Common values for \( y \): \( -1, 0, 1 \) (3 values).

#### For \( x = 2 \):
1. From the circle:
\[ y^2 < 16 - 2^2 \Rightarrow y^2 < 12 \Rightarrow -\sqrt{12} < y < \sqrt{12} \]
The integral values of \( y \) are \( -3, -2, -1, 0, 1, 2, 3 \) (7 values).

2. From the parabola:
\[ y^2 < 4 \cdot 2 \Rightarrow y^2 < 8 \Rightarrow -\sqrt{8} < y < \sqrt{8} \]
The integral values of \( y \) are \( -2, -1, 0, 1, 2 \) (5 values).

3. Common values for \( y \): \( -2, -1, 0, 1, 2 \) (5 values).

#### For \( x = 3 \):
1. From the circle:
\[ y^2 < 16 - 3^2 \Rightarrow y^2 < 7 \Rightarrow -\sqrt{7} < y < \sqrt{7} \]
The integral values of \( y \) are \( -2, -1, 0, 1, 2 \) (5 values).

2. From the parabola:
\[ y^2 < 4 \cdot 3 \Rightarrow y^2 < 12 \Rightarrow -\sqrt{12} < y < \sqrt{12} \]
The integral values of \( y \) are \( -3, -2, -1, 0, 1, 2, 3 \) (7 values).

3. Common values for \( y \): \( -2, -1, 0, 1, 2 \) (5 values).

### Step 5: Total integral points
Now we can sum up the number of integral points:
- For \( x = 1 \): 3 values of \( y \)
- For \( x = 2 \): 5 values of \( y \)
- For \( x = 3 \): 5 values of \( y \)

Total integral points = \( 3 + 5 + 5 = 13 \).

### Final Answer
The number of points with integral coordinates that lie in the interior of the region common to the circle and the parabola is **13**.

---