A line through `A(-5,-4)` meets the lines `x+3y+2=0,2x+y+4=0 and x-y-5=0` at the points `B , Ca n dD` rspectively, if `((15)/(A B))^2+((10)/(A C))^2=(6/(A D))^2` find the equation of the line.

Explanation

To find the equation of the line passing through the point \( A(-5, -4) \) that meets the lines \( x + 3y + 2 = 0 \), \( 2x + y + 4 = 0 \), and \( x - y - 5 = 0 \) at points \( B \), \( C \), and \( D \) respectively, we can follow these steps:

### Step 1: Define the Line through Point A
Let the slope of the line through point \( A \) be \( m \). The equation of the line in point-slope form is:
\[ y + 4 = m(x + 5) \]
This can be rearranged to:
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