Consider the parabola whose focus is at (0,0) and tangent at vertex is x-y+1=0
The length of the chord of parabola on the x-axis is

Options

A. `4sqrt(2)`
B. `2sqrt(2)`
C. `8sqrt(2)`
D. `3sqrt(2)`

Explanation

To solve the problem, we need to find the length of the chord of the parabola on the x-axis given the focus and the tangent at the vertex. Let's break down the solution step by step.

### Step 1: Identify the focus and tangent
The focus of the parabola is given as \( (0, 0) \) and the equation of the tangent at the vertex is \( x - y + 1 = 0 \).

### Step 2: Find the distance from the focus to the tangent
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by the formula:

\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For our tangent line \( x - y + 1 = 0 \), we have \( A = 1, B = -1, C = 1 \) and the focus \( (x_0, y_0) = (0, 0) \).

Substituting these values into the distance formula:

\[ d = \frac{|1 \cdot 0 + (-1) \cdot 0 + 1|}{\sqrt{1^2 + (-1)^2}} = \frac{|1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 3: Find the distance between the focus and the directrix
The distance from the focus to the directrix is \( 2d \):

\[ 2d = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \]

### Step 4: Determine the equation of the directrix
Since the directrix is parallel to the tangent, its equation can be written as \( x - y + \lambda = 0 \). The distance from the focus \( (0, 0) \) to the directrix must equal \( \sqrt{2} \):

\[ \frac{|0 - 0 + \lambda|}{\sqrt{1^2 + (-1)^2}} = \sqrt{2} \]

This simplifies to:

\[ \frac{|\lambda|}{\sqrt{2}} = \sqrt{2} \implies |\lambda| = 2 \]

Thus, \( \lambda = 2 \) (since we can take the positive value for the directrix). Therefore, the equation of the directrix is:

\[ x - y + 2 = 0 \]

### Step 5: Set up the parabola's equation
Using the definition of a parabola (the set of points equidistant from the focus and the directrix), we can set up the equation. Let \( P(x, y) \) be a point on the parabola. The distance from \( P \) to the focus \( (0, 0) \) is:

\[ \sqrt{x^2 + y^2} \]

The distance from \( P \) to the directrix \( x - y + 2 = 0 \) is:

\[ \frac{|x - y + 2|}{\sqrt{2}} \]

Setting these distances equal gives:

\[ \sqrt{x^2 + y^2} = \frac{|x - y + 2|}{\sqrt{2}} \]

Squaring both sides and simplifying leads to the equation of the parabola.

### Step 6: Find the intersection with the x-axis
To find the length of the chord on the x-axis, we set \( y = 0 \) in the parabola's equation and solve for \( x \):

1. Substitute \( y = 0 \) into the parabola's equation derived in the previous step.
2. This will yield a quadratic equation in \( x \).

### Step 7: Solve the quadratic equation
Assuming we derived the quadratic equation as \( x^2 - 4x - 4 = 0 \):

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ x = \frac{4 \pm \sqrt{16 + 16}}{2} = \frac{4 \pm \sqrt{32}}{2} = \frac{4 \pm 4\sqrt{2}}{2} = 2 \pm 2\sqrt{2} \]

### Step 8: Calculate the length of the chord
The x-coordinates of the intersection points are \( 2 + 2\sqrt{2} \) and \( 2 - 2\sqrt{2} \). The length of the chord is:

\[ \text{Length} = |(2 + 2\sqrt{2}) - (2 - 2\sqrt{2})| = |4\sqrt{2}| \]

Thus, the length of the chord of the parabola on the x-axis is \( 4\sqrt{2} \).

### Final Answer:
The length of the chord of the parabola on the x-axis is \( 4\sqrt{2} \).

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