A point P lies on the line L₀ : x+1 2 = y 1 = z-5 0 such that its x -coordinate is positive. If the distance of the line L₁ : x-1 2 = y-1 1 = z-2 1 from the point P , measured along a direction parallel to the vector 2 i + j - k , is 2 6 , then the sum of the coordinates of P is :

Step-by-step solution

Let the point P on the line L₀ be (2s-1, s, 5) . The line passing through P and parallel to the vector 2 i + j - k has the parametric equations: x = 2s - 1 + 2t y = s + t z = 5 - t Let this line intersect L₁ at a point Q . Since Q lies on L₁ , its coordinates must satisfy the equation of L₁ : 2s - 1 + 2t - 1 2 = s + t - 1 1 = 5 - t - 2 1 From the first two parts: 2s + 2t - 2 2 = s + t - 1 s + t - 1 = s + t - 1 (which is an identity, meaning the first two coordinate ratios are consistent). From the last two parts: s