The points of intersection of the parabolas `y^(2) = 5x` and `x^(2) = 5y` lie on the line

Options

A. x+y=10
B. x-2y=0
C. x-y=0
D. 2x-y=0

Answer

To find the points of intersection of the parabolas \(y^2 = 5x\) and \(x^2 = 5y\), we will follow these steps:

### Step 1: Write down the equations of the parabolas
We have two equations:
1. \(y^2 = 5x\) (Equation 1)
2. \(x^2 = 5y\) (Equation 2)

### Step 2: Express \(y\) in terms of \(x\) from Equation 1
From Equation 1, we can express \(y\) as:
\[ y = \sqrt{5x} \quad \text{(considering the positive root for now)} \]

### Step 3: Substitute \(y\) into Equation 2
Now, substitute \(y\) from Equation 1 into Equation 2:
\[ x^2 = 5(\sqrt{5x}) \]
This simplifies to:
\[ x^2 = 5\sqrt{5x} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ x^4 = 25 \cdot 5x \]
\[ x^4 = 125x \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ x^4 - 125x = 0 \]

### Step 6: Factor out \(x\)
Factoring out \(x\) results in:
\[ x(x^3 - 125) = 0 \]

### Step 7: Solve for \(x\)
This gives us two cases:
1. \(x = 0\)
2. \(x^3 - 125 = 0\) which simplifies to \(x^3 = 125\) or \(x = 5\)

### Step 8: Find corresponding \(y\) values
Now, we will find the corresponding \(y\) values for \(x = 0\) and \(x = 5\):
- For \(x = 0\):
\[ y^2 = 5(0) \Rightarrow y = 0 \]
So, one point of intersection is \((0, 0)\).

- For \(x = 5\):
\[ y^2 = 5(5) \Rightarrow y^2 = 25 \Rightarrow y = 5 \quad \text{(considering the positive root)} \]
So, another point of intersection is \((5, 5)\).

### Step 9: Identify the line through the points
The points of intersection are \((0, 0)\) and \((5, 5)\). The line passing through these points can be described by the equation:
\[ y - 0 = \frac{5 - 0}{5 - 0}(x - 0) \Rightarrow y = x \]
This can be rearranged to:
\[ x - y = 0 \]

### Conclusion
The points of intersection of the parabolas lie on the line \(x - y = 0\).