If parabola of latus rectum l touches a fixed equal parabola, the axes of the two curves being parallel, then the locus of the vertex of the moving curve is

Answer

To solve the problem step by step, we will find the locus of the vertex of a moving parabola that touches a fixed parabola, given that both parabolas have parallel axes.

### Step-by-Step Solution:

1. **Identify the Fixed Parabola**:
We start with the fixed parabola, which we can express as:
\[ y^2 = 4ax \]
This is a standard form of a parabola that opens to the right.

2. **Define the Moving Parabola**:
The moving parabola, which is equal and has parallel axes, can be represented as:
\[ (y - \beta)^2 = -4a(x - \alpha) \]
Here, \(\alpha\) and \(\beta\) represent the coordinates of the vertex of the moving parabola.

3. **Substitute for x**:
Since the moving parabola touches the fixed parabola, we will substitute \(x\) from the fixed parabola's equation into the moving parabola's equation. From the fixed parabola, we have:
\[ x = \frac{y^2}{4a} \]
Substituting this into the equation of the moving parabola gives:
\[ (y - \beta)^2 = -4a\left(\frac{y^2}{4a} - \alpha\right) \]

4. **Simplify the Equation**:
Simplifying this, we get:
\[ (y - \beta)^2 = -y^2 + 4a\alpha \]
Expanding the left side:
\[ y^2 - 2y\beta + \beta^2 = -y^2 + 4a\alpha \]
Rearranging gives:
\[ 2y^2 - 2y\beta + \beta^2 - 4a\alpha = 0 \]

5. **Condition for Tangency**:
For the two parabolas to touch, the discriminant of this quadratic equation must be zero:
\[ D = b^2 - 4ac = 0 \]
Here, \(b = -2\beta\), \(a = 2\), and \(c = \beta^2 - 4a\alpha\). Thus:
\[ (-2\beta)^2 - 4(2)(\beta^2 - 4a\alpha) = 0 \]
Simplifying this gives:
\[ 4\beta^2 - 8(\beta^2 - 4a\alpha) = 0 \]
\[ 4\beta^2 - 8\beta^2 + 32a\alpha = 0 \]
\[ -4\beta^2 + 32a\alpha = 0 \]

6. **Finding the Locus**:
Rearranging the above equation gives:
\[ 32a\alpha = 4\beta^2 \]
Dividing through by 4:
\[ 8a\alpha = \beta^2 \]
This can be expressed as:
\[ \beta^2 = 8a\alpha \]

7. **Final Locus Equation**:
The locus of the vertex \((\alpha, \beta)\) can be written as:
\[ y^2 = 8ax \]
This is the required locus of the vertex of the moving parabola.

### Final Answer:
The locus of the vertex of the moving parabola is:
\[ y^2 = 8ax \]