If `(-2,5)` and (3,7) are the points of intersection of the tangent and normal at a point on a parabola with the axis of the parabola, then the focal distance of that point is

Options

A. `(sqrt(29))/(2)`
B. `(5)/(2)`
C. `sqrt(29)`
D. `(2)/(5)`

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the Given Information
We are given two points of intersection of the tangent and normal at a point on a parabola. The points are \((-2, 5)\) and \((3, 7)\).

### Step 2: Write the Equation of the Parabola
The standard form of the parabola with its axis along the x-axis is given by:
\[ y^2 = 4ax \]
where \(a\) is the distance from the vertex to the focus.

### Step 3: Identify the Point on the Parabola
Let the point on the parabola be \(P(at^2, 2at)\), where \(t\) is the parameter.

### Step 4: Write the Equation of the Tangent and Normal
The equation of the tangent at point \(P\) is given by:
\[ ty = x + at^2 \]
The equation of the normal at point \(P\) is given by:
\[ y - 2at = -\frac{1}{t}(x - at^2) \]

### Step 5: Find the Midpoint of the Tangent and Normal
The midpoint \(M\) of the segment joining the points of intersection of the tangent and normal can be calculated using the coordinates of the given points:
\[ M = \left( \frac{-2 + 3}{2}, \frac{5 + 7}{2} \right) = \left( \frac{1}{2}, 6 \right) \]

### Step 6: Calculate the Distance \(TN\)
The distance \(TN\) between the points \((-2, 5)\) and \((3, 7)\) can be calculated using the distance formula:
\[ TN = \sqrt{(3 - (-2))^2 + (7 - 5)^2} = \sqrt{(3 + 2)^2 + (2)^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \]

### Step 7: Relate the Focal Distance to the Distance \(TN\)
According to the properties of the parabola, the focal distance \(PF\) is given by:
\[ PF = \frac{TN}{2} \]
Thus,
\[ PF = \frac{\sqrt{29}}{2} \]

### Step 8: Conclusion
The focal distance of the point on the parabola is:
\[ \frac{\sqrt{29}}{2} \]