The number of the tangents that can be drawn from (1, 2) to `x^(2)+y^(2)=5`, is

Options

A. 1
B. 2
C. 3
D. 0

Answer

To determine the number of tangents that can be drawn from the point (1, 2) to the circle defined by the equation \(x^2 + y^2 = 5\), we can follow these steps:

### Step 1: Identify the Circle's Center and Radius
The given equation of the circle is \(x^2 + y^2 = 5\).
- The center of the circle is at the origin (0, 0).
- The radius \(r\) can be found as follows:
\[ r = \sqrt{5} \]

### Step 2: Calculate the Distance from the Point to the Center of the Circle
Next, we need to find the distance from the point (1, 2) to the center of the circle (0, 0). The distance \(d\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 3: Compare the Distance with the Radius
Now, we compare the distance \(d\) with the radius \(r\):
- We have \(d = \sqrt{5}\) and \(r = \sqrt{5}\).

### Step 4: Determine the Position of the Point Relative to the Circle
According to the properties of tangents:
- If \(d < r\), the point lies inside the circle, and the number of tangents is 0.
- If \(d = r\), the point lies on the circle, and the number of tangents is 1.
- If \(d > r\), the point lies outside the circle, and the number of tangents is 2.

Since \(d = r\), we conclude that the point (1, 2) lies on the circle.

### Step 5: Conclusion
Since the point lies on the circle, the number of tangents that can be drawn from the point (1, 2) to the circle \(x^2 + y^2 = 5\) is:
\[ \text{Number of tangents} = 1 \]

### Final Answer
The number of tangents that can be drawn from the point (1, 2) to the circle \(x^2 + y^2 = 5\) is **1**.

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