The equation of the directrix of the parabola with vertex at the origin and having the axis along the x-axis and a common tangent of slope 2 with the circle `x^2+y^2=5` is (are) `x=10` (b) `x=20` `x=-10` (d) `x=-20`

Options

A. x=10
B. x=20
C. x=-10
D. x=-20

Explanation

To solve the problem, we need to find the equation of the directrix of a parabola with a vertex at the origin, axis along the x-axis, and a common tangent of slope 2 with the circle \(x^2 + y^2 = 5\).

### Step-by-Step Solution:

1. **Identify the Parabola's Equation**:
The parabola with vertex at the origin and axis along the x-axis can be represented as:
\[ y^2 = 4ax \]
where \(a\) is the distance from the vertex to the focus.

2. **Equation of the Circle**:
The given circle is:
\[ x^2 + y^2 = 5 \]
This means the radius \(r\) of the circle is \(\sqrt{5}\).

3. **Equation of the Tangent to the Circle**:
The general equation of the tangent to the circle \(x^2 + y^2 = r^2\) is given by:
\[ y = mx \pm \sqrt{r^2(1 + m^2)} \]
Here, the slope \(m = 2\) and \(r^2 = 5\). Thus, the equation becomes:
\[ y = 2x \pm \sqrt{5(1 + 2^2)} = 2x \pm \sqrt{5 \cdot 5} = 2x \pm 5 \]
Therefore, the tangents to the circle are:
\[ y = 2x + 5 \quad \text{and} \quad y = 2x - 5 \]

4. **Equation of the Tangent to the Parabola**:
The equation of the tangent to the parabola \(y^2 = 4ax\) is given by:
\[ y = mx + \frac{a}{m} \]
Substituting \(m = 2\):
\[ y = 2x + \frac{a}{2} \]

5. **Setting the Tangents Equal**:
Since the tangent is common to both the circle and the parabola, we can equate the two expressions for \(y\):
\[ 2x + \frac{a}{2} = 2x + 5 \]
This simplifies to:
\[ \frac{a}{2} = 5 \implies a = 10 \]

6. **Finding the Directrix**:
The directrix of the parabola \(y^2 = 4ax\) is given by the equation:
\[ x = -a \]
Substituting \(a = 10\):
\[ x = -10 \]

7. **Considering the Other Parabola**:
Since the parabola can also open to the left, we can also consider:
\[ x = a \implies x = 10 \]

### Final Answer:
Thus, the equations of the directrix for the two possible parabolas are:
- \(x = -10\)
- \(x = 10\)

The correct options from the given choices are:
- (c) \(x = -10\)
- (a) \(x = 10\)