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Let the point P be the intersection of the line L₁: x-1 2 = y+1 1 = z-2 1 and the plane ₁: x + 2y + 2z = 15 . Let L₂ be a line passing through P and perpendicular to the plane ₁ . If Q is the foot of the perpendicular from the point R(4, 5, 5) to the line L₂ and is the area of the triangle PQR , then the value of 4 ^2 is equal to _____ .

Correct answer

81

Step-by-step solution

Any point on the line L₁ is given by (2 +1, -1, +2) . Since P is the intersection of L₁ and ₁ , it must satisfy the plane's equation x + 2y + 2z = 15 : (2 +1) + 2( -1) + 2( +2) = 15 2 + 1 + 2 - 2 + 2 + 4 = 15 6 + 3 = 15 = 2 Thus, the coordinates of P are (5, 1, 4) . The line L₂ passes through P and is perpendicular to ₁ . Its direction ratios are proportional to the normal vector of ₁ , which is 1, 2, 2 . The equation of L₂ is x-5 1 = y-1 2 = z-4 2 = . Any point on L₂ is S( +5, 2 +1, 2 +4) . The vector connecting R

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