The number of tangents that can be drawn from the point (0,1) to the circle `x^(2)+y^(2)-2x -4y=0` is

Options

A. 0
B. 1
C. 2
D. none

Answer

To solve the problem of finding the number of tangents that can be drawn from the point (0,1) to the circle given by the equation \(x^2 + y^2 - 2x - 4y = 0\), we will follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we need to rewrite the equation of the circle in standard form. The given equation is:

\[ x^2 + y^2 - 2x - 4y = 0 \]

We can rearrange this to complete the square for both \(x\) and \(y\).

### Step 2: Completing the Square
For \(x\):
\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]

For \(y\):
\[ y^2 - 4y \rightarrow (y - 2)^2 - 4 \]

Now substituting back into the equation:

\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 = 0 \]

This simplifies to:

\[ (x - 1)^2 + (y - 2)^2 - 5 = 0 \]

Thus, we have:

\[ (x - 1)^2 + (y - 2)^2 = 5 \]

### Step 3: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center: \((h, k) = (1, 2)\)
- Radius: \(r = \sqrt{5}\)

### Step 4: Calculate the Distance from the Point to the Center
Next, we need to calculate the distance from the point (0, 1) to the center of the circle (1, 2):

Using the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates:

\[ d = \sqrt{(1 - 0)^2 + (2 - 1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 5: Compare the Distance with the Radius
Now, we compare the distance \(d\) with the radius \(r\):

- Distance \(d = \sqrt{2}\)
- Radius \(r = \sqrt{5}\)

Since \(d < r\), the point (0, 1) lies inside the circle.

### Step 6: Conclusion about the Number of Tangents
From the properties of circles, we know that:
- If a point is inside the circle, no tangents can be drawn from that point to the circle.
- If the point is on the circle, exactly one tangent can be drawn.
- If the point is outside the circle, two tangents can be drawn.

Since (0, 1) is inside the circle, we conclude that:

**The number of tangents that can be drawn from the point (0, 1) to the circle is 0.**

### Final Answer
The number of tangents is **0**.

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