The locus of the image of the point `(2,3)` in the line `(x-2y+3)+lambda(2x-3y+4)=0` is`(lambda in R)` (a) `x^2+y^2-3x-4y-4=0` (b) `2x^2+3y^2+2x+4y-7=0` (c) `x^2+y^2-2x-4y+4=0` (d) none of these

Explanation

To find the locus of the image of the point (2, 3) in the line given by the equation \( (x - 2y + 3) + \lambda(2x - 3y + 4) = 0 \), we can follow these steps:

### Step 1: Write the equations of the lines
The given equation can be split into two lines:
1. \( L_1: x - 2y + 3 = 0 \)
2. \( L_2: 2x - 3y + 4 = 0 \)

### Step 2: Solve for the intersection of the lines
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