Let B and C are points of interection of the parabola `y=x^(2)` and the circle `x^(2)+(y-2)^(2)=8.` The area of the triangle OBC, where O is the origin, is

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the area of triangle OBC, where O is the origin and B and C are the points of intersection of the parabola \( y = x^2 \) and the circle \( x^2 + (y - 2)^2 = 8 \), we will follow these steps:

### Step 1: Set up the equations
We have the equations of the parabola and the circle:
- Parabola: \( y = x^2 \)
- Circle: \( x^2 + (y - 2)^2 = 8 \)

### Step 2: Substitute the parabola equation into the circle equation
Substituting \( y = x^2 \) into the circle equation:
\[ x^2 + (x^2 - 2)^2 = 8 \]
Expanding the equation:
\[ x^2 + (x^4 - 4x^2 + 4) = 8 \]
This simplifies to:
\[ x^4 - 3x^2 - 4 = 0 \]

### Step 3: Let \( z = x^2 \) and solve the quadratic equation
Let \( z = x^2 \). Then we have:
\[ z^2 - 3z - 4 = 0 \]
Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ z = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \]
Calculating the roots:
\[ z = 4 \quad \text{and} \quad z = -1 \]
Since \( z = x^2 \), we discard \( z = -1 \) as it is not valid. Thus, \( z = 4 \).

### Step 4: Find the corresponding x-values
From \( z = 4 \), we have:
\[ x^2 = 4 \implies x = \pm 2 \]
Now substituting back to find \( y \):
\[ y = x^2 = 4 \]
Thus, the points of intersection B and C are:
- \( B(-2, 4) \)
- \( C(2, 4) \)

### Step 5: Find the area of triangle OBC
The vertices of the triangle OBC are:
- \( O(0, 0) \)
- \( B(-2, 4) \)
- \( C(2, 4) \)

To find the area of triangle OBC, we can use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base BC is the distance between points B and C:
\[ \text{Base} = |x_C - x_B| = |2 - (-2)| = 4 \]
The height is the y-coordinate of points B and C since they are both at \( y = 4 \) and the origin is at \( y = 0 \):
\[ \text{Height} = 4 \]

### Step 6: Calculate the area
Now substituting the base and height into the area formula:
\[ \text{Area} = \frac{1}{2} \times 4 \times 4 = \frac{16}{2} = 8 \]

### Final Answer
The area of triangle OBC is \( 8 \) square units.