To determine the circle to which two tangents can be drawn from the origin, we will analyze the given equations of circles step by step.
### Step 1: Understand the Circle Equation
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this equation, we can identify the center of the circle as \((-g, -f)\) and the radius \(r\) can be calculated using:
\[ r = \sqrt{g^2 + f^2 - c} \]
### Step 2: Analyze the First Circle
The first circle is given by:
\[ x^2 + y^2 - 8x - 4y - 3 = 0 \]
**Identify g, f, and c:**
- \(g = -8\)
- \(f = -4\)
- \(c = -3\)
**Calculate the center and radius:**
- Center: \((4, 2)\) (since \(-g, -f\))
- Radius:
\[
r = \sqrt{(-8)^2 + (-4)^2 - (-3)} = \sqrt{64 + 16 + 3} = \sqrt{83}
\]
**Calculate distance from origin to center (OC):**
\[
OC = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}
\]
**Check the condition for tangents:**
Since \(OC < r\) (i.e., \(\sqrt{20} < \sqrt{83}\)), there are no tangents from the origin to this circle.
### Step 3: Analyze the Second Circle
The second circle is given by:
\[ x^2 + y^2 + 4x + 2y - 2 = 0 \]
**Identify g, f, and c:**
- \(g = 4\)
- \(f = 2\)
- \(c = -2\)
**Calculate the center and radius:**
- Center: \((-4, -2)\)
- Radius:
\[
r = \sqrt{(4)^2 + (2)^2 - (-2)} = \sqrt{16 + 4 + 2} = \sqrt{22}
\]
**Calculate distance from origin to center (OC):**
\[
OC = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}
\]
**Check the condition for tangents:**
Since \(OC > r\) (i.e., \(\sqrt{20} > \sqrt{22}\)), there are two tangents from the origin to this circle.
### Step 4: Analyze the Third Circle
The third circle is given by:
\[ x^2 + y^2 - 8x + 6y - 1 = 0 \]
**Identify g, f, and c:**
- \(g = -8\)
- \(f = 6\)
- \(c = -1\)
**Calculate the center and radius:**
- Center: \((4, -3)\)
- Radius:
\[
r = \sqrt{(-8)^2 + (6)^2 - (-1)} = \sqrt{64 + 36 + 1} = \sqrt{101}
\]
**Calculate distance from origin to center (OC):**
\[
OC = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
**Check the condition for tangents:**
Since \(OC > r\) (i.e., \(5 > \sqrt{101}\)), there are two tangents from the origin to this circle.
### Conclusion
From the analysis, the circles from the second and third equations allow for two tangents to be drawn from the origin.
### Final Answer
The circles to which two tangents can be drawn from the origin are the second and third circles.
