The two parabolas `y^(2)=4x" and "x^(2)=4y` intersect at a point P, whose abscissas is not zero, such that

Options

A. they both touch each other at P
B. they cut at right angles at P
C. the tangents to each curvs at P make complementary angles with the x-axis
D. none of these

Answer

To solve the problem of finding the intersection points of the parabolas \( y^2 = 4x \) and \( x^2 = 4y \), we will follow these steps:

### Step 1: Set the equations equal to each other
We have two equations:
1. \( y^2 = 4x \)
2. \( x^2 = 4y \)

To find the points of intersection, we can express \( y \) in terms of \( x \) from the first equation and substitute it into the second equation.

### Step 2: Substitute \( y \) from the first equation into the second
From \( y^2 = 4x \), we can express \( y \) as:
\[ y = \sqrt{4x} = 2\sqrt{x} \]
Now, substitute \( y \) into the second equation \( x^2 = 4y \):
\[ x^2 = 4(2\sqrt{x}) \implies x^2 = 8\sqrt{x} \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ x^2 - 8\sqrt{x} = 0 \]
Let \( \sqrt{x} = t \), then \( x = t^2 \). Substituting \( t \) into the equation gives:
\[ (t^2)^2 - 8t = 0 \implies t^4 - 8t = 0 \]
Factoring out \( t \):
\[ t(t^3 - 8) = 0 \]

### Step 4: Solve for \( t \)
Setting each factor to zero gives:
1. \( t = 0 \) (which corresponds to \( x = 0 \))
2. \( t^3 - 8 = 0 \implies t^3 = 8 \implies t = 2 \) (which corresponds to \( x = 4 \))

### Step 5: Find the corresponding \( y \) values
For \( t = 2 \):
\[ \sqrt{x} = 2 \implies x = 4 \]
Now substitute \( x = 4 \) back into the first equation to find \( y \):
\[ y^2 = 4(4) = 16 \implies y = 4 \text{ or } y = -4 \]
Thus, the points of intersection are:
1. \( (0, 0) \) (which we discard since the abscissa is zero)
2. \( (4, 4) \)
3. \( (4, -4) \)

### Step 6: Verify the conditions
The problem states that we need to find the point \( P \) whose abscissa is not zero. Therefore, the valid points are \( (4, 4) \) and \( (4, -4) \).

### Conclusion
The two parabolas intersect at the points \( (4, 4) \) and \( (4, -4) \), where the abscissa is not zero.

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