If tangents PA and PB are drawn from `P(-1, 2)` to `y^(2) = 4x` then

Options

A. Equation of AB is `y = x -1`
B. Length of AB is 8
C. Length of AB is 4
D. Equation of AB is `y = x + 1 `

Answer

To find the length of the chord of contact (AB) from the point P(-1, 2) to the parabola \( y^2 = 4x \), we can follow these steps:

### Step 1: Write the equation of the chord of contact
The equation of the chord of contact from a point \( (x_1, y_1) \) to the parabola \( y^2 = 4ax \) is given by:
\[ yy_1 = 2a(x + x_1) \]
For the parabola \( y^2 = 4x \), we have \( a = 1 \). Therefore, the equation becomes:
\[ yy_1 = 2(x + x_1) \]
Substituting \( P(-1, 2) \) into the equation where \( x_1 = -1 \) and \( y_1 = 2 \):
\[ y \cdot 2 = 2(x - 1) \]
This simplifies to:
\[ 2y = 2(x - 1) \]
Dividing both sides by 2:
\[ y = x - 1 \]

### Step 2: Find the points of intersection of the line with the parabola
Now, we need to find the points where the line \( y = x - 1 \) intersects the parabola \( y^2 = 4x \). We substitute \( y = x - 1 \) into the parabola's equation:
\[ (x - 1)^2 = 4x \]
Expanding this gives:
\[ x^2 - 2x + 1 = 4x \]
Rearranging the equation:
\[ x^2 - 6x + 1 = 0 \]

### Step 3: Solve the quadratic equation
To find the roots of the quadratic equation \( x^2 - 6x + 1 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -6, c = 1 \):
\[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{6 \pm \sqrt{36 - 4}}{2} = \frac{6 \pm \sqrt{32}}{2} = \frac{6 \pm 4\sqrt{2}}{2} = 3 \pm 2\sqrt{2} \]
Thus, the x-coordinates of the points of intersection are:
\[ x_1 = 3 + 2\sqrt{2}, \quad x_2 = 3 - 2\sqrt{2} \]

### Step 4: Find the corresponding y-coordinates
Now we find the y-coordinates using \( y = x - 1 \):
\[ y_1 = (3 + 2\sqrt{2}) - 1 = 2 + 2\sqrt{2}, \quad y_2 = (3 - 2\sqrt{2}) - 1 = 2 - 2\sqrt{2} \]

### Step 5: Calculate the length of the chord AB
The length of the chord AB can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ AB = \sqrt{((3 - 2\sqrt{2}) - (3 + 2\sqrt{2}))^2 + ((2 - 2\sqrt{2}) - (2 + 2\sqrt{2}))^2} \]
Calculating the differences:
\[ = \sqrt{((-4\sqrt{2})^2) + ((-4\sqrt{2})^2)} = \sqrt{32 + 32} = \sqrt{64} = 8 \]

### Final Answer
The length of the chord AB is \( 8 \).

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