Two parabola have the same focus. If their directrices are the x-axis and the y-axis respectively, then the slope of their common chord is :

Options

A. -1
B. `-1//2`
C. `-sqrt(3)//2`
D. none of these

Explanation

To solve the problem, we need to find the equations of the two parabolas and then determine the slope of their common chord. Here’s a step-by-step solution:

### Step 1: Understand the Parabolas
We have two parabolas with the same focus (A, B) and different directrices: one is the x-axis and the other is the y-axis.

### Step 2: Equation of the First Parabola
For the parabola with the focus (A, B) and directrix as the x-axis, we use the definition of a parabola:
\[ PF = PM \]
Where:
- \( PF \) is the distance from point \( P(x, y) \) to the focus \( (A, B) \).
- \( PM \) is the distance from point \( P(x, y) \) to the directrix (x-axis), which is simply \( y \).

Calculating \( PF \):
\[ PF = \sqrt{(x - A)^2 + (y - B)^2} \]
Calculating \( PM \):
\[ PM = y \]

Setting these equal gives:
\[ \sqrt{(x - A)^2 + (y - B)^2} = y \]

### Step 3: Square Both Sides
Squaring both sides to eliminate the square root:
\[ (x - A)^2 + (y - B)^2 = y^2 \]

### Step 4: Simplify the Equation
Expanding the left side:
\[ (x - A)^2 + y^2 - 2By + B^2 = y^2 \]
Cancelling \( y^2 \) from both sides:
\[ (x - A)^2 - 2By + B^2 = 0 \]

### Step 5: Rearranging the First Parabola's Equation
Rearranging gives:
\[ (x - A)^2 + B^2 = 2By \]

### Step 6: Equation of the Second Parabola
For the parabola with the focus (A, B) and directrix as the y-axis, we again use the definition:
\[ PF = PM \]
Where \( PM \) is the distance from point \( P(x, y) \) to the directrix (y-axis), which is simply \( x \).

Calculating \( PF \) (same as before):
\[ PF = \sqrt{(x - A)^2 + (y - B)^2} \]
Calculating \( PM \):
\[ PM = x \]

Setting these equal gives:
\[ \sqrt{(x - A)^2 + (y - B)^2} = x \]

### Step 7: Square Both Sides Again
Squaring both sides:
\[ (x - A)^2 + (y - B)^2 = x^2 \]

### Step 8: Simplify the Second Parabola's Equation
Expanding the left side:
\[ (x - A)^2 + y^2 - 2By + B^2 = x^2 \]
Cancelling \( x^2 \) from both sides:
\[ (x - A)^2 - x^2 - 2By + B^2 = 0 \]

### Step 9: Rearranging the Second Parabola's Equation
Rearranging gives:
\[ -A^2 + 2Ax + B^2 = 2By \]

### Step 10: Finding the Common Chord
To find the common chord, we subtract the two equations:
\[ [(x - A)^2 + B^2 - 2By] - [(-A^2 + 2Ax + B^2 - 2By)] = 0 \]

This simplifies to:
\[ y^2 - x^2 = 0 \]

### Step 11: Factor the Equation
Factoring gives:
\[ (y - x)(y + x) = 0 \]

### Step 12: Finding the Slope
This results in two lines:
1. \( y = x \) (slope = 1)
2. \( y = -x \) (slope = -1)

Since we are looking for the slope of the common chord, we take the negative slope:
\[ \text{Slope} = -1 \]

### Final Answer
The slope of the common chord is \( -1 \).