If a circle passes through the points of intersection of the co-ordinate axes with the lines `lambda x-y+1=0 and x-2y+3=0`, then the value of `lambda` is

Options

A. 2
B. `1//3`
C. 6
D. 3

Answer

To solve the problem, we need to find the value of \( \lambda \) such that a circle passes through the points of intersection of the coordinate axes with the lines given by \( \lambda x - y + 1 = 0 \) and \( x - 2y + 3 = 0 \).

### Step-by-Step Solution:

1. **Find the intersection points of the first line with the axes:**
The equation of the first line is:
\[ \lambda x - y + 1 = 0 \]
- To find the x-intercept, set \( y = 0 \):
\[ \lambda x + 1 = 0 \implies x = -\frac{1}{\lambda} \]
So the x-intercept is \( A\left(-\frac{1}{\lambda}, 0\right) \).
- To find the y-intercept, set \( x = 0 \):
\[ -y + 1 = 0 \implies y = 1 \]
So the y-intercept is \( B(0, 1) \).

2. **Find the intersection points of the second line with the axes:**
The equation of the second line is:
\[ x - 2y + 3 = 0 \]
- To find the x-intercept, set \( y = 0 \):
\[ x + 3 = 0 \implies x = -3 \]
So the x-intercept is \( C(-3, 0) \).
- To find the y-intercept, set \( x = 0 \):
\[ -2y + 3 = 0 \implies y = \frac{3}{2} \]
So the y-intercept is \( D\left(0, \frac{3}{2}\right) \).

3. **Determine the condition for concyclicity:**
The points \( A, B, C, D \) are concyclic if the following condition holds:
\[ OA \cdot OC = OB \cdot OD \]
where \( O \) is the origin.

4. **Calculate the distances:**
- Distance \( OA \) (from origin to point A):
\[ OA = \sqrt{\left(-\frac{1}{\lambda}\right)^2 + 0^2} = \frac{1}{|\lambda|} \]
- Distance \( OB \) (from origin to point B):
\[ OB = \sqrt{0^2 + 1^2} = 1 \]
- Distance \( OC \) (from origin to point C):
\[ OC = \sqrt{(-3)^2 + 0^2} = 3 \]
- Distance \( OD \) (from origin to point D):
\[ OD = \sqrt{0^2 + \left(\frac{3}{2}\right)^2} = \frac{3}{2} \]

5. **Set up the equation:**
Substitute the distances into the concyclicity condition:
\[ OA \cdot OC = OB \cdot OD \]
This gives:
\[ \frac{1}{|\lambda|} \cdot 3 = 1 \cdot \frac{3}{2} \]
Simplifying this, we have:
\[ \frac{3}{|\lambda|} = \frac{3}{2} \]

6. **Solve for \( \lambda \):**
Cross-multiplying yields:
\[ 3 \cdot 2 = 3 \cdot |\lambda| \implies 6 = 3|\lambda| \implies |\lambda| = 2 \]
Thus, \( \lambda = 2 \) or \( \lambda = -2 \). However, since we are looking for a specific value, we take \( \lambda = 2 \).

### Final Answer:
The value of \( \lambda \) is \( 2 \).