Area bounded by the parabola `y = x^(2) - 2x + 3` and tangents drawn to it from the point P(1, 0) is equal to

Options

A. `4sqrt(2)` sq. units
B. `(4sqrt(2))/(3)` sq. units
C. `(8sqrt(2))/(3)` sq. units
D. `(16sqrt(2))/(3)` sq. units

Answer

To find the area bounded by the parabola \( y = x^2 - 2x + 3 \) and the tangents drawn from the point \( P(1, 0) \), we will follow these steps:

### Step 1: Rewrite the Parabola
The given parabola can be rewritten in vertex form:
\[ y = (x - 1)^2 + 2 \]
This indicates that the vertex of the parabola is at the point \( (1, 2) \).

### Step 2: Determine the Tangents from Point P
To find the tangents from the point \( P(1, 0) \), we will use the formula for the tangent line to a parabola \( y = ax^2 + bx + c \) at a point \( (x_0, y_0) \):
\[ y - y_0 = m(x - x_0) \]
where \( m \) is the slope of the tangent line.

### Step 3: Find the Slope of the Tangent
The slope \( m \) can be found using the condition that the line must satisfy the equation of the parabola. We can derive the equation of the tangent line at a point \( (x_1, y_1) \) on the parabola:
\[ y - (x_1^2 - 2x_1 + 3) = m(x - x_1) \]

### Step 4: Set Up the Equation
Substituting \( P(1, 0) \) into the tangent equation gives:
\[ 0 - (x_1^2 - 2x_1 + 3) = m(1 - x_1) \]
This leads to:
\[ x_1^2 - 2x_1 + 3 = -m(1 - x_1) \]

### Step 5: Find Points of Tangency
The tangents from point \( P(1, 0) \) will intersect the parabola at two points. We can find the values of \( x_1 \) that satisfy the quadratic equation formed by substituting \( y = mx + c \) into the parabola's equation.

### Step 6: Solve for the Area
The area between the parabola and the tangents can be computed using the integral:
\[ \text{Area} = 2 \int_{1}^{1 + \sqrt{2}} \left( (x^2 - 2x + 3) - (mx + c) \right) dx \]
where \( m \) and \( c \) are determined from the tangents.

### Step 7: Evaluate the Integral
Now, we will evaluate the integral:
1. Calculate the definite integral from \( 1 \) to \( 1 + \sqrt{2} \).
2. Multiply the result by 2 to account for both sides of the parabola.

### Final Calculation
After computing the integral and simplifying, we find that the area bounded by the parabola and the tangents is:
\[ \text{Area} = \frac{4\sqrt{2}}{3} \]