Let be the line of intersection of the planes given by the equationsLet be the line passing through the point and parallel to . Let denote the plane given by the equationSuppose that the line meets the plane at the point . Let be the foot of the perpendicular drawn from to the plane .If the length of the line segment is and the area of is , find the value of 4$$\ell$$^2 A^2.\
[THOUGHT] I need to compute the length ℓ = |PQ| and the area A of triangle PQR, then evaluate 4ℓ²A².
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Find direction vector of L₁ as the cross product of the normals of the two planes: n₁ = (2,3,1), n₂ = (1,2,1) → d = n₁ × n₂ = (1, -1, 1).
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Parameterize L₂ through P(2, -1, 3) with direction d: L₂: (x,y,z) = (2, -1, 3) + t·(1, -1, 1) = (2+t, -1-t, 3+t).
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Find Q = L₂ ∩ Π by solving 2x + y - 2z = 6 for t: Plug L₂ into Π → -3 - t = 6 → t = -9. So Q = (-7, 8, -6).
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Compute ℓ = |PQ| = sqrt((Δx)² + (Δy)² + (Δz)²): Δx = -9, Δy = 9, Δz = -9 → ℓ = sqrt(243) = 9√3.
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Find foot R of perpendicular from P to Π: Normal of Π: n = (2,1,-2). Use a point on Π, e.g., (0,0,-3). Signed distance = n·(P - (0,0,-3))/||n|| = -9/3 = -3. R = P - signed_distance·(n/||n||) = P + n = (4,0,1).
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Compute area of triangle PQR: Vectors: v₁ = Q - P = (-9,9,-9), v₂ = R - P = (2,1,-2). Cross product v₁ × v₂ = (-9, -36, -27), magnitude = sqrt(2106) = 9√26. Area A = ½·|cross| = (9/2)√26.
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Evaluate 4ℓ²A²: ℓ² = 243, A² = (81/4)·26 = 1053/2. 4ℓ²A² = 4·243·(1053/2) = 486·1053 = 511758.
All steps are exact; I will verify with SymPy. [/THOUGHT]
[PYTHON] import sympy as sp
Define symbols
t = sp.symbols('t') P = sp.Matrix([2, -1, 3]) n1 = sp.Matrix([2, 3, 1]) n2 = sp.Matrix([1, 2, 1]) d = n1.cross(n2) # direction vector of L1
Parameterize L2
L2 = P + t * d
Find Q: intersection with plane Pi: 2x + y - 2z = 6
Pi_eq = sp.Eq(2L2[0] + L2[1] - 2L2[2], 6) t_val = sp.solve(Pi_eq, t)[0] Q = L2.subs(t, t_val)
Length ℓ = |PQ|
ell = sp.sqrt((Q - P).dot(Q - P))
Find foot R of perpendicular from P to Pi
n = sp.Matrix([2, 1, -2]) # normal of Pi
Choose a point on Pi: set x=0, y=0 → -2z = 6 → z = -3
A0 = sp.Matrix([0, 0, -3]) signed_dist = n.dot(P - A0) / n.norm() R = P - signed_dist * n / n.norm()
Area of triangle PQR
v1 = Q - P v2 = R - P cross = v1.cross(v2) area = sp.Rational(1, 2) * sp.sqrt(cross.dot(cross))
Compute 4ℓ^2A^2
result = 4 * ell2 * area2 print(result.simplify()) [/PYTHON]
[THOUGHT] The Python code verified the calculations step-by-step, confirming that the value of (4\ell^2 A^2) is 511758. This result is consistent with the geometric computations involving the direction vector of the line, intersection with the plane, and the area of the triangle formed by the perpendicular foot. [/THOUGHT]
\boxed{511758}