The parabola `y^2 = 4x` and the circle `(x-6)^2 + y^2 = r^2` will have no common tangent if r is equal to

Options

A. `rgtsqrt20`
B. `rltsqrt20`
C. `rgtsqrt18`
D. `rin(sqrt20,sqrt28)`

Answer

To find the value of \( r \) such that the parabola \( y^2 = 4x \) and the circle \( (x - 6)^2 + y^2 = r^2 \) have no common tangent, we can follow these steps:

### Step 1: Identify the Parabola and Circle
The equation of the parabola is given as:
\[ y^2 = 4x \]
This is a standard parabola that opens to the right.

The equation of the circle is given as:
\[ (x - 6)^2 + y^2 = r^2 \]
This circle is centered at the point \( (6, 0) \).

### Step 2: Find the Normal to the Parabola
The normal to the parabola at a point \( (t^2, 2t) \) can be expressed as:
\[ y - 2t = -tx + t^2 \]
Rearranging gives us:
\[ tx + y - 2t - t^2 = 0 \]

### Step 3: Substitute the Center of the Circle
Since the normal passes through the center of the circle \( (6, 0) \), we substitute \( x = 6 \) and \( y = 0 \) into the normal equation:
\[ t(6) + 0 - 2t - t^2 = 0 \]
This simplifies to:
\[ 6t - 2t - t^2 = 0 \implies t^2 - 4t = 0 \]

### Step 4: Factor the Equation
Factoring gives:
\[ t(t - 4) = 0 \]
Thus, \( t = 0 \) or \( t = 4 \).

### Step 5: Find Points on the Parabola
For \( t = 4 \):
\[ P = (t^2, 2t) = (16, 8) \]
For \( t = 0 \):
\[ P = (0, 0) \]

### Step 6: Calculate the Distance from the Center to the Point
Now, we calculate the distance \( CP \) from the center of the circle \( C(6, 0) \) to the point \( P(16, 8) \):
\[ CP = \sqrt{(16 - 6)^2 + (8 - 0)^2} = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} = 2\sqrt{41} \]

### Step 7: Determine Conditions for No Common Tangent
For the circle and the parabola to have no common tangents, the radius \( r \) must be less than the distance \( CP \):
\[ r < 2\sqrt{41} \]

### Conclusion
Thus, the value of \( r \) for which the parabola and the circle have no common tangent is:
\[ r < 2\sqrt{41} \]