The tangents at the end points of any chord through `(1, 0)` to the parabola `y^2 + 4x = 8` intersect

Options

A. `"at "45^(@)" on "x-3=0`
B. `"at "45^(@)" on "x+3=0`
C. `"at "90^(@)" on "x+3=0`
D. `"at "90^(@)" on "x-3=0`

Explanation

To solve the problem step by step, let's follow the reasoning provided in the video transcript.

### Step 1: Convert the parabola to standard form
The given equation of the parabola is:
\[ y^2 + 4x = 8 \]

To convert it to standard form, we rearrange the equation:
\[ y^2 = 8 - 4x \]
\[ y^2 = -4(x - 2) \]

This can be rewritten as:
\[ (y - 0)^2 = -4(x - 2) \]

This is now in the standard form of a parabola, which is:
\[ (y - k)^2 = 4p(x - h) \]
where the vertex is at \((h, k)\) and \(p\) is the distance from the vertex to the focus.

### Step 2: Identify the vertex and focus
From the standard form \((y - 0)^2 = -4(x - 2)\), we can identify:
- Vertex: \((h, k) = (2, 0)\)
- The value of \(p\) is \(-1\) (since \(4p = -4\)), indicating that the parabola opens to the left.

The focus of the parabola is located at:
\[ (h + p, k) = (2 - 1, 0) = (1, 0) \]

### Step 3: Find the equation of the directrix
The directrix of the parabola can be found using the formula \(x = h - p\):
\[ x = 2 + 1 = 3 \]

Thus, the equation of the directrix is:
\[ x = 3 \]

### Step 4: Determine the properties of the tangents
According to the properties of parabolas, the tangents at the endpoints of any chord through the focus (in this case, the point \((1, 0)\)) are perpendicular to each other and intersect at the directrix.

### Step 5: Conclusion
Thus, the tangents at the endpoints of any chord through the point \((1, 0)\) to the parabola \(y^2 + 4x = 8\) will intersect at the directrix, which is given by the line:
\[ x = 3 \]

### Final Answer
The tangents at the endpoints of any chord through \((1, 0)\) to the parabola \(y^2 + 4x = 8\) intersect at the line \(x = 3\).

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