If y=3x is a tangent to a circle with centre (1,1) then the other tangent drawn through (0,0) to the circle is

Options

A. `3y=x`
B. `y=-3x`
C. `y=2x`
D. `3y=-2x`

Answer

To find the other tangent drawn through the point (0,0) to the circle centered at (1,1) with the given tangent line \( y = 3x \), we can follow these steps:

### Step 1: Identify the center of the circle and the radius
The center of the circle is given as \( (1, 1) \). The line \( y = 3x \) is a tangent to the circle.

### Step 2: Find the distance from the center to the tangent line
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( y = 3x \), we can rewrite it in the form \( Ax + By + C = 0 \) as \( 3x - y = 0 \). Here, \( A = 3 \), \( B = -1 \), and \( C = 0 \).

Substituting the center \( (1, 1) \) into the distance formula:
\[ d = \frac{|3(1) - 1(1) + 0|}{\sqrt{3^2 + (-1)^2}} = \frac{|3 - 1|}{\sqrt{9 + 1}} = \frac{2}{\sqrt{10}} = \frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5} \]

### Step 3: Set the radius equal to the distance
Since the distance from the center to the tangent line is equal to the radius \( r \) of the circle, we have:
\[ r = \frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5} \]

### Step 4: Write the equation of the other tangent line
Let the equation of the other tangent line through the origin be \( y = mx \). The distance from the center \( (1, 1) \) to this line can be calculated similarly:
\[ d = \frac{|m(1) - 1(1)|}{\sqrt{m^2 + 1}} = \frac{|m - 1|}{\sqrt{m^2 + 1}} \]
Setting this equal to the radius:
\[ \frac{|m - 1|}{\sqrt{m^2 + 1}} = \frac{2\sqrt{10}}{10} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (m - 1)^2 = \frac{4 \cdot 10}{100}(m^2 + 1) \]
\[ (m - 1)^2 = \frac{2}{5}(m^2 + 1) \]

### Step 6: Expand and rearrange the equation
Expanding both sides:
\[ m^2 - 2m + 1 = \frac{2}{5}m^2 + \frac{2}{5} \]
Multiplying through by 5 to eliminate the fraction:
\[ 5m^2 - 10m + 5 = 2m^2 + 2 \]
Rearranging gives:
\[ 3m^2 - 10m + 3 = 0 \]

### Step 7: Factor the quadratic equation
Factoring the quadratic:
\[ (3m - 1)(m - 3) = 0 \]
Thus, the solutions for \( m \) are:
\[ m = \frac{1}{3} \quad \text{or} \quad m = 3 \]

### Step 8: Write the equations of the tangents
The equations of the tangents are:
1. \( y = 3x \) (given)
2. \( y = \frac{1}{3}x \)

### Final Answer
The other tangent drawn through (0,0) to the circle is:
\[ y = \frac{1}{3}x \]