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. `pm1`
B. `4//3`
C. `3//4`
D. none of these

Explanation

To solve the problem, we need to find the slope of the common chord of two parabolas that have the same focus but different directrices. Let's break down the solution step by step.

### Step 1: Define the Focus and Directrices
Let the focus of both parabolas be \( F(a, b) \). The directrix of the first parabola is the x-axis (which can be represented as \( y = 0 \)), and the directrix of the second parabola is the y-axis (which can be represented as \( x = 0 \)).

### Step 2: Write the Equation of the First Parabola
Using the definition of a parabola, the distance from any point \( P(x, y) \) on the parabola to the focus \( F(a, b) \) is equal to the distance from \( P \) to the directrix.

For the first parabola (directrix is the x-axis):
\[ PF = \sqrt{(x - a)^2 + (y - b)^2} \]
The distance from point \( P \) to the x-axis (directrix) is simply \( y \).

Setting these distances equal gives:
\[ \sqrt{(x - a)^2 + (y - b)^2} = y \]
Squaring both sides:
\[ (x - a)^2 + (y - b)^2 = y^2 \]

### Step 3: Simplify the Equation of the First Parabola
Expanding and simplifying:
\[ (x - a)^2 + (y^2 - 2by + b^2) = y^2 \]
This simplifies to:
\[ (x - a)^2 + b^2 - 2by = 0 \]
Rearranging gives the equation of the first parabola:
\[ (x - a)^2 + b^2 = 2by \]

### Step 4: Write the Equation of the Second Parabola
For the second parabola (directrix is the y-axis):
\[ PF = \sqrt{(x - a)^2 + (y - b)^2} \]
The distance from point \( P \) to the y-axis (directrix) is \( x \).

Setting these distances equal gives:
\[ \sqrt{(x - a)^2 + (y - b)^2} = x \]
Squaring both sides:
\[ (x - a)^2 + (y - b)^2 = x^2 \]

### Step 5: Simplify the Equation of the Second Parabola
Expanding and simplifying:
\[ (x - a)^2 + (y^2 - 2by + b^2) = x^2 \]
This simplifies to:
\[ (x - a)^2 + b^2 - 2by = x^2 \]
Rearranging gives the equation of the second parabola:
\[ (x - a)^2 + b^2 = 2bx \]

### Step 6: Find the Common Chord
To find the common chord, we subtract the two equations derived from the parabolas:
\[ [(x - a)^2 + b^2 - 2by] - [(x - a)^2 + b^2 - 2bx] = 0 \]
This simplifies to:
\[ -2by + 2bx = 0 \]
Factoring out \( 2 \):
\[ 2(bx - by) = 0 \]
Thus, we have:
\[ bx = by \]
This implies:
\[ y = x \]
and
\[ y = -x \]

### Step 7: Determine the Slope
The equations \( y = x \) and \( y = -x \) represent lines with slopes of \( 1 \) and \( -1 \), respectively. Therefore, the slopes of the common chord are:
\[ \text{slope} = \pm 1 \]

### Final Answer
The slope of their common chord is \( \pm 1 \).

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