A family of lines is given by `(1+2lambda)x+(1-lambda)y+lambda=0` being a parameter . The line belonging to this family at the maximum distance from the point (1,4) is `ax+by+c=0` , then find the value of `(a+b+c)/16`

Options

A. `33x+12y+7=0`
B. `12x+33y-7=0`
C. `4x-y+7=0`
D. `12x-33y+7=0`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Rewrite the Family of Lines
The family of lines is given by the equation:
\[ (1 + 2\lambda)x + (1 - \lambda)y + \lambda = 0 \]
We can rewrite this in a more standard form:
\[ (1 + 2\lambda)x + (1 - \lambda)y + \lambda = 0 \]