A circle of radius `r,r ne 0` touches the parabola `y ^(2) + 12 x =0` at the ertex of the parabola. The centre of the circle lies to the left of the vertex and the circle lies entriely within the parabola. Then the intervals in which, r lies can be

Options

A. `(1,7)`
B. `(0,6)`
C. `(1, (13)/(2))`
D. `(2,8)`

Answer

To solve the problem, we need to find the intervals in which the radius \( r \) of the circle lies, given that the circle touches the parabola \( y^2 + 12x = 0 \) at its vertex and lies entirely within the parabola.

### Step-by-Step Solution:

1. **Identify the Parabola**:
The given equation of the parabola is \( y^2 + 12x = 0 \). This can be rewritten as \( y^2 = -12x \), which indicates that it opens to the left. The vertex of the parabola is at the origin \( (0, 0) \).

**Hint**: Remember that the vertex of a parabola in the form \( y^2 = 4px \) is at the point \( (0, 0) \) when \( p \) is negative.

2. **Position of the Circle**:
The circle touches the parabola at the vertex, which is \( (0, 0) \). Since the center of the circle lies to the left of the vertex, we can denote the center of the circle as \( (-r, 0) \), where \( r \) is the radius of the circle.

**Hint**: The center of the circle is positioned at \( (-r, 0) \) because it is to the left of the vertex.

3. **Equation of the Circle**:
The equation of the circle can be expressed as:
\[ (x + r)^2 + y^2 = r^2 \]

**Hint**: Remember the standard form of the circle's equation centered at \( (h, k) \) is \( (x - h)^2 + (y - k)^2 = r^2 \).

4. **Finding Intersection Points**:
To find where the circle intersects the parabola, substitute \( y^2 = -12x \) into the circle's equation:
\[ (x + r)^2 + (-12x) = r^2 \]
Simplifying this gives:
\[ (x + r)^2 - 12x = r^2 \]
Expanding and rearranging:
\[ x^2 + 2rx + r^2 - 12x - r^2 = 0 \]
This simplifies to:
\[ x^2 + (2r - 12)x = 0 \]

**Hint**: Factor out \( x \) from the equation to find the roots.

5. **Finding Roots**:
Factoring gives:
\[ x(x + 2r - 12) = 0 \]
The roots are \( x = 0 \) and \( x = 12 - 2r \).

**Hint**: The circle touches the parabola at \( (0, 0) \), so we need to ensure that the second root \( 12 - 2r \) is non-positive for the circle to lie entirely within the parabola.

6. **Condition for the Circle to Lie Within the Parabola**:
For the circle to lie entirely within the parabola, we need:
\[ 12 - 2r \leq 0 \]
Solving this inequality gives:
\[ 2r \geq 12 \quad \Rightarrow \quad r \geq 6 \]

**Hint**: Remember that the radius \( r \) must also be positive.

7. **Combine Conditions**:
Since \( r \) must be positive and less than 6 (as derived from the conditions of the circle touching the parabola), we combine the conditions:
\[ 0 < r < 6 \]

**Final Result**:
The interval in which \( r \) lies is:
\[ r \in (0, 6) \]

### Summary of the Intervals:
The radius \( r \) must be in the interval \( (0, 6) \) for the circle to touch the parabola at the vertex and lie entirely within it.