The focus of a parabola is (0, 0) and vertex (1, 1). If two mutually perpendicular tangents can be drawn to a parabola from the circle `(x-2)^(2)+(y-3)^(2)=r^(2)`,then

Options

A. `|r|gt1/sqrt2`
B. `rgt1/sqrt2`
C. `r=1/sqrt2`
D. `|r|lt1/sqrt2`

Answer

To solve the problem step by step, we need to find the equation of the parabola, the equation of the directrix, and then analyze the conditions under which the circle intersects the directrix.

### Step 1: Identify the focus and vertex of the parabola
The focus of the parabola is given as \( F(0, 0) \) and the vertex is given as \( V(1, 1) \).

### Step 2: Find the equation of the parabola
The standard form of a parabola that opens towards the focus is given by the formula:
\[ (y - k) = \frac{1}{4p}(x - h)^2 \]
where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.

In this case:
- Vertex \( V(1, 1) \)
- Focus \( F(0, 0) \)

The distance \( p \) can be calculated as:
\[ p = \sqrt{(1 - 0)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]

Since the focus is below the vertex, the parabola opens downwards. The equation becomes:
\[ (y - 1) = -\frac{1}{4\sqrt{2}}(x - 1)^2 \]

### Step 3: Find the equation of the directrix
The directrix of a parabola is given by the formula:
\[ y = k + p \]
For our case:
\[ y = 1 + \sqrt{2} \]

### Step 4: Analyze the circle
The circle is given by the equation:
\[ (x - 2)^2 + (y - 3)^2 = r^2 \]
The center of the circle is \( C(2, 3) \).

### Step 5: Find the distance from the center of the circle to the directrix
The equation of the directrix is \( y = 1 + \sqrt{2} \). The distance \( D \) from the center of the circle to the directrix can be calculated using the formula for the distance from a point to a line:
\[ D = \frac{|y_0 - (1 + \sqrt{2})|}{\sqrt{1^2}} = |3 - (1 + \sqrt{2})| = |2 - \sqrt{2}| \]

### Step 6: Determine the condition for the circle to intersect the directrix
For the circle to have two mutually perpendicular tangents to the parabola, the distance \( D \) must be less than the radius \( r \):
\[ |2 - \sqrt{2}| < r \]

### Conclusion
Thus, the condition for the circle to intersect the directrix at two points, allowing for two mutually perpendicular tangents to be drawn from the circle to the parabola, is:
\[ r > |2 - \sqrt{2}| \]