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

Options

A. `-1`
B. `-1/2`
C. `4//3`
D. `3//4`

Answer

To solve the problem, we need to find the slope of the common chord of two parabolas that share the same focus but have different directrices. The directrices are the x-axis and the y-axis, respectively.

### Step-by-Step Solution:

1. **Identify the Focus and Directrices**:
Let the focus of both parabolas be \( F(a, b) \).
- The first parabola has its directrix as the x-axis (y = 0).
- The second parabola has its directrix as the y-axis (x = 0).

2. **Equation of the First Parabola**:
According to the definition of a parabola, the distance from a point \( P(x, y) \) to the focus \( F(a, b) \) is equal to the distance from \( P \) to the directrix.
- For the first parabola (directrix y = 0):
\[ PF = PM \]
where \( PM \) is the distance from \( P \) to the x-axis (y = 0).
\[ \sqrt{(x - a)^2 + (y - b)^2} = y \]
- Squaring both sides:
\[ (x - a)^2 + (y - b)^2 = y^2 \]
- Expanding and simplifying:
\[ (x - a)^2 + y^2 - 2by + b^2 = y^2 \]
\[ (x - a)^2 + b^2 - 2by = 0 \]
\[ (x - a)^2 + b^2 = 2by \]
- Rearranging gives:
\[ y = \frac{(x - a)^2 + b^2}{2b} \quad \text{(Equation 1)} \]

3. **Equation of the Second Parabola**:
- For the second parabola (directrix x = 0):
\[ PF = PM \]
where \( PM \) is the distance from \( P \) to the y-axis (x = 0).
\[ \sqrt{(x - a)^2 + (y - b)^2} = x \]
- Squaring both sides:
\[ (x - a)^2 + (y - b)^2 = x^2 \]
- Expanding and simplifying:
\[ (x - a)^2 + y^2 - 2by + b^2 = x^2 \]
\[ (x - a)^2 + y^2 - x^2 + b^2 - 2by = 0 \]
\[ (x - a)^2 + b^2 = x^2 + 2by \]
- Rearranging gives:
\[ y = \frac{(x - a)^2 + b^2 - x^2}{2b} \quad \text{(Equation 2)} \]

4. **Finding the Common Chord**:
To find the common chord, we need to eliminate \( y \) from both equations. We can do this by setting the right-hand sides of both equations equal to each other:
\[ \frac{(x - a)^2 + b^2}{2b} = \frac{(x - a)^2 + b^2 - x^2}{2b} \]
- Cross-multiplying and simplifying leads to:
\[ (x - a)^2 + b^2 = (x - a)^2 + b^2 - x^2 \]
- This simplifies to:
\[ x^2 = (x - a)^2 \]
- Expanding gives:
\[ x^2 = x^2 - 2ax + a^2 \]
- Rearranging leads to:
\[ 2ax = a^2 \quad \Rightarrow \quad x = \frac{a}{2} \quad \text{(for } a \neq 0\text{)} \]

5. **Finding the Slope**:
Substitute \( x = \frac{a}{2} \) back into either equation to find \( y \):
\[ y = \frac{(x - a)^2 + b^2}{2b} \]
- The slope of the common chord can be determined from the equations \( y = x \) and \( y = -x \), which gives us slopes of \( 1 \) and \( -1 \).

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