If the tangents at two points `(1,2) and (3,6)` on a parabola intersect at the point `(-1,1),` then ths slope of the directrix of the parabola is

Options

A. `sqrt2`
B. `-2`
C. `-1`
D. none of these

Answer

To find the slope of the directrix of the parabola given the points and tangents, we can follow these steps:

### Step 1: Identify the Points and Tangent Equations
We are given two points on the parabola: \( P_1(1, 2) \) and \( P_2(3, 6) \). The tangents at these points intersect at the point \( (-1, 1) \).

### Step 2: Write the General Equation of the Parabola
Assuming the parabola is in the form \( y^2 = 4ax \), we can express the points in terms of the parameters of the parabola. The general equation of the parabola can also be expressed in vertex form as \( (y - k)^2 = 4a(x - h) \).

### Step 3: Find the Tangent Equations
The equation of the tangent to the parabola at a point \( (x_1, y_1) \) is given by:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the tangent at that point.

For point \( P_1(1, 2) \):
- The slope \( m_1 \) can be found using the derivative of the parabola.

For point \( P_2(3, 6) \):
- Similarly, find the slope \( m_2 \).

### Step 4: Set Up the Tangent Equations
Using the slopes and the points, we can write the equations of the tangents at \( P_1 \) and \( P_2 \):
1. Tangent at \( P_1(1, 2) \):
\[ y - 2 = m_1(x - 1) \]
2. Tangent at \( P_2(3, 6) \):
\[ y - 6 = m_2(x - 3) \]

### Step 5: Solve for the Intersection Point
Since both tangents intersect at \( (-1, 1) \), we can substitute \( x = -1 \) and \( y = 1 \) into both equations to find \( m_1 \) and \( m_2 \).

### Step 6: Calculate the Slopes
From the tangent equations, substituting \( (-1, 1) \):
1. For the first tangent:
\[ 1 - 2 = m_1(-1 - 1) \implies -1 = -2m_1 \implies m_1 = \frac{1}{2} \]
2. For the second tangent:
\[ 1 - 6 = m_2(-1 - 3) \implies -5 = -4m_2 \implies m_2 = \frac{5}{4} \]

### Step 7: Find the Slope of the Directrix
The slope of the directrix of a parabola is given by the negative reciprocal of the slope of the axis of the parabola. The axis of symmetry can be found from the average of the slopes of the tangents:
\[ m_{avg} = \frac{m_1 + m_2}{2} = \frac{\frac{1}{2} + \frac{5}{4}}{2} = \frac{\frac{2 + 5}{4}}{2} = \frac{7/4}{2} = \frac{7}{8} \]
The slope of the directrix is then:
\[ m_{directrix} = -\frac{1}{m_{avg}} = -\frac{8}{7} \]

### Final Answer
The slope of the directrix of the parabola is \( -\frac{8}{7} \).