Let `y^2=4ax` be a parabola and `x^2+y^2+2bx=0` be a circle. If parabola and circle touch each externally then:

Options

A. `agt0,blt0`
B. `agt0,bgt0`
C. `alt0,bgt0`
D. `alt0,blt0`

Answer

To determine the conditions under which the parabola \(y^2 = 4ax\) and the circle \(x^2 + y^2 + 2bx = 0\) touch each other externally, we will analyze the equations step by step.

### Step 1: Understand the equations
The given parabola is \(y^2 = 4ax\), which opens to the right. The standard form indicates that \(a > 0\) for the parabola to open rightwards.

The circle's equation can be rewritten as:
\[ x^2 + y^2 = -2bx \]
This can be rearranged to:
\[ x^2 + 2bx + y^2 = 0 \]
Completing the square for \(x\):
\[ (x + b)^2 + y^2 = b^2 \]
This shows that the circle is centered at \((-b, 0)\) with a radius of \(|b|\).

### Step 2: Analyze the conditions for external tangency
For the parabola and circle to touch externally, the distance between the center of the circle and the vertex of the parabola must equal the radius of the circle.

1. **Vertex of the parabola**: The vertex of the parabola \(y^2 = 4ax\) is at the origin \((0, 0)\).
2. **Center of the circle**: The center of the circle is at \((-b, 0)\).
3. **Distance between the vertex and the center**: The distance \(d\) between the vertex \((0, 0)\) and the center \((-b, 0)\) is:
\[ d = |0 - (-b)| = |b| \]
4. **Radius of the circle**: The radius of the circle is \(|b|\).

### Step 3: Set up the condition for tangency
For the circle to touch the parabola externally, we need:
\[ d = \text{radius} \]
Thus:
\[ |b| = |b| \]
This condition is trivially satisfied. However, we need to consider the cases based on the signs of \(a\) and \(b\).

### Step 4: Analyze different cases
1. **Case 1**: \(a > 0\) and \(b > 0\)
- The parabola opens to the right, and the circle is centered in the negative x-direction. They can touch externally.

2. **Case 2**: \(a > 0\) and \(b < 0\)
- The parabola still opens to the right, and the circle is centered in the positive x-direction. They can also touch externally.

3. **Case 3**: \(a < 0\) and \(b > 0\)
- The parabola opens to the left. The circle can be positioned such that they touch externally.

4. **Case 4**: \(a < 0\) and \(b < 0\)
- The parabola opens to the left, and the circle is centered in the positive x-direction. They can still touch externally.

### Conclusion
Thus, both \(a\) and \(b\) can take on positive or negative values under certain conditions, and the parabola and circle can touch each other externally in all cases.