Two equal parabolas have the same vertex and their axes are at right angles. The length of the common tangent to them, is

Options

A. 3a
B. `3sqrt2a`
C. 6a
D. 2a

Explanation

To find the length of the common tangent to two equal parabolas that have the same vertex and their axes at right angles, we can follow these steps:

### Step 1: Define the Parabolas
We consider the two parabolas:
1. \( y^2 = 4ax \) (horizontal parabola)
2. \( x^2 = 4ay \) (vertical parabola)

### Step 2: Equation of the Tangent to the First Parabola
The equation of the tangent to the first parabola \( y^2 = 4ax \) can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent.

### Step 3: Find the Point of Contact on the First Parabola
The point of contact \( P \) on the first parabola can be determined as follows:
- From the tangent equation, substituting \( y = mx + \frac{a}{m} \) into the parabola equation gives the coordinates of the point of contact:
\[ P = \left( \frac{a}{m^2}, \frac{2a}{m} \right) \]

### Step 4: Equation of the Tangent to the Second Parabola
For the second parabola \( x^2 = 4ay \), the equation of the tangent can be expressed as:
\[ x = my + \frac{a}{m} \]

### Step 5: Find the Point of Contact on the Second Parabola
The point of contact \( Q \) on the second parabola can be determined similarly:
- From the tangent equation, substituting \( x = my + \frac{a}{m} \) into the parabola equation gives the coordinates of the point of contact:
\[ Q = \left( 2am, \frac{a}{m^2} \right) \]

### Step 6: Condition for Common Tangent
For the tangent to be common to both parabolas, the points of contact \( P \) and \( Q \) must satisfy the condition:
\[ \frac{a}{m^2} = 2am \quad \text{and} \quad \frac{2a}{m} = \frac{a}{m^2} \]
From the first equation, we can derive:
\[ \frac{1}{m^2} = 2m \implies m^3 = \frac{1}{2} \implies m = -1 \]

### Step 7: Coordinates of Points of Contact
Substituting \( m = -1 \) back into the coordinates:
- For point \( P \):
\[ P = \left( a, -2a \right) \]
- For point \( Q \):
\[ Q = \left( -2a, a \right) \]

### Step 8: Calculate the Length of the Common Tangent
The length of the common tangent \( PQ \) can be calculated using the distance formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( P \) and \( Q \):
\[ PQ = \sqrt{(-2a - a)^2 + (a - (-2a))^2} \]
\[ = \sqrt{(-3a)^2 + (3a)^2} = \sqrt{9a^2 + 9a^2} = \sqrt{18a^2} = 3\sqrt{2}a \]

### Final Answer
Thus, the length of the common tangent to the two parabolas is:
\[ \boxed{3\sqrt{2}a} \]