The area (in square units) of the triangle bounded by x = 4 and the lines `y^(2)-x^(2)+2x=1` is equal to

Options

A. 3
B. 6
C. 12
D. 9

Answer

To find the area of the triangle bounded by the line \( x = 4 \) and the curve \( y^2 - x^2 + 2x = 1 \), we can follow these steps:

### Step 1: Rewrite the given equation
The equation of the curve is given as:
\[ y^2 - x^2 + 2x = 1 \]
We can rearrange this to express \( y^2 \):
\[ y^2 = x^2 - 2x + 1 \]
This simplifies to:
\[ y^2 = (x - 1)^2 \]

### Step 2: Find the expressions for \( y \)
Taking the square root, we find:
\[ y = x - 1 \quad \text{and} \quad y = -(x - 1) \]
Thus, the two lines are:
\[ y = x - 1 \quad \text{(Line 1)} \quad \text{and} \quad y = -x + 1 \quad \text{(Line 2)} \]

### Step 3: Identify the points of intersection
Next, we need to find the points where these lines intersect the line \( x = 4 \).

For \( y = x - 1 \):
\[ y = 4 - 1 = 3 \quad \Rightarrow \quad (4, 3) \]

For \( y = -x + 1 \):
\[ y = -4 + 1 = -3 \quad \Rightarrow \quad (4, -3) \]

### Step 4: Find the point of intersection of the two lines
To find the intersection of the two lines \( y = x - 1 \) and \( y = -x + 1 \):
\[ x - 1 = -x + 1 \]
Solving for \( x \):
\[ 2x = 2 \quad \Rightarrow \quad x = 1 \]
Substituting \( x = 1 \) into either equation:
\[ y = 1 - 1 = 0 \quad \Rightarrow \quad (1, 0) \]

### Step 5: Determine the vertices of the triangle
The vertices of the triangle are:
1. \( (4, 3) \)
2. \( (4, -3) \)
3. \( (1, 0) \)

### Step 6: Calculate the area of the triangle
The area \( A \) of a triangle given vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
\[ A = \frac{1}{2} \left| 4(-3 - 0) + 4(0 - 3) + 1(3 - (-3)) \right| \]
Calculating:
\[ A = \frac{1}{2} \left| 4(-3) + 4(-3) + 1(6) \right| \]
\[ = \frac{1}{2} \left| -12 - 12 + 6 \right| = \frac{1}{2} \left| -18 \right| = \frac{18}{2} = 9 \]

### Final Answer
The area of the triangle is \( 9 \) square units.

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