If $3x+y=0$ is a tangent to the circle with centre (2, 1), then the equation of another tangent to the same circle from origin is:

Question

Accepted Answer

To determine why Option 3 ($x - 3y = 0$) is the correct answer, we need to analyze the problem and the properties of tangents to a circle.

### Explanation:

1. **Understanding the Given Tangent:**
   - The equation of the given tangent is $3x + y = 0$.
   - The circle has its center at (2, 1).
   - The distance from the center of the circle to the tangent line can be calculated using the formula for the distance from a point to a line: 
     $$
     \text{Distance} = \frac{|3(2) + 1(1) + 0|}{\sqrt{3^2 + 1^2}} = \frac{|6 + 1|}{\sqrt{10}} = \frac{7}{\sqrt{10}}
     $$
   - This distance represents the radius of the circle.

2. **Finding Another Tangent from the Origin:**
   - A tangent from the origin to the circle will have the same radius, $\frac{7}{\sqrt{10}}$.
   - The line $x - 3y = 0$ passes through the origin and is perpendicular to the line $3x + y = 0$.
   - The slope of $3x + y = 0$ is $-3$, so the perpendicular slope is $\frac{1}{3}$, which matches the slope of $x - 3y = 0$.

3. **Verification of Option 3:**
   - The line $x - 3y = 0$ has a slope of $\frac{1}{3}$, which is perpendicular to the slope of the given tangent $3x + y = 0$.
   - Since tangents from a point to a circle are perpendicular to the radius at the point of tangency, $x - 3y = 0$ is indeed a valid tangent from the origin.

### Why Other Options are Incorrect:

1. **Option 1 ($x + 3y = 0$):**
   - The slope of this line is $-1/3$, which is not perpendicular to the slope of $3x + y = 0$.
   - Therefore, it cannot be a tangent from the origin.

2. **Option 2 ($3x - y = 0$):**
   - The slope of this line is $3$, which is the same as the negative reciprocal of the slope of the given tangent.
   - However, it does not satisfy the condition of being perpendicular to the radius at the point of tangency from the origin.

3. **Option 4 ($x + y = 0$):**
   - The slope of this line is $-1$, which is not perpendicular to the slope of $3x + y = 0$.
   - Thus, it cannot be a tangent from the origin.

In conclusion, Option 3 ($x - 3y = 0$) is the correct answer because it satisfies the condition of being a tangent from the origin to the circle, with the correct perpendicular relationship to the given tangent line.