If the perpendicular distance of the line `lx+my =1` from the point (0, 0) is a, then the diameter of the circle touching the given line and with centre (0, 0) is

Options

A. 2a units
B. a units
C. `(a)/(2)` units
D. 3a units

Answer

To solve the problem, we need to find the diameter of a circle centered at the origin (0, 0) that touches the line given by the equation \( lx + my = 1 \). The perpendicular distance from the origin to this line is given as \( a \).

### Step-by-Step Solution:

1. **Understand the Perpendicular Distance Formula**:
The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to a line given by \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
In our case, the line can be rewritten as \( lx + my - 1 = 0 \), where \( A = l \), \( B = m \), and \( C = -1 \).

2. **Apply the Formula**:
For the point \( (0, 0) \):
\[ d = \frac{|l(0) + m(0) - 1|}{\sqrt{l^2 + m^2}} = \frac{|-1|}{\sqrt{l^2 + m^2}} = \frac{1}{\sqrt{l^2 + m^2}} \]
We know from the problem statement that this distance is equal to \( a \):
\[ a = \frac{1}{\sqrt{l^2 + m^2}} \]

3. **Find the Radius of the Circle**:
Since the circle touches the line, the radius of the circle is equal to the perpendicular distance from the center (0, 0) to the line, which we have found to be \( a \).

4. **Calculate the Diameter**:
The diameter \( D \) of a circle is given by:
\[ D = 2 \times \text{radius} \]
Therefore, substituting the radius:
\[ D = 2a \]

### Final Answer:
The diameter of the circle is \( 2a \) units.