<p>If the image of the point \(P(1, -2, 3)\) in the plane, \(2x + 3y - 4z + 22 = 0\) measured parallel to the line, \(\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}\) is \(Q\), then \(PQ\) is equal to</p>
Step-by-Step Solution
Key Concept: The image Q of point P in a plane measured parallel to a line is found by moving P along the line's direction until reaching the plane, then reflecting that intersection point. The distance PQ equals twice the perpendicular distance from P to the plane along the given direction.
Step 1: Write the parametric equation of line through P(1, -2, 3) parallel to the given direction (1, 4, 5): x = 1 + t, y = -2 + 4t, z = 3 + 5t Step 2: Find where this line meets the plane 2x + 3y - 4z + 22 = 0: 2(1+t) + 3(-2+4t) - 4(3+5t) + 22 = 0 2 + 2t - 6 + 12t - 12 - 20t + 22 = 0 -6t + 6 = 0 t = 1 Step 3: Find the foot of perpendicular M where line intersects plane: M = (1+1, -2+4, 3+5) = (2, 2, 8) Step 4: Since Q is the image of P measured parallel to the line, M is the midpoint of PQ. Therefore Q = 2M - P: Q = 2(2, 2, 8) - (1, -2, 3) = (4, 4, 16) - (1, -2, 3) = (3, 6, 13) Step 5: Calculate PQ: PQ = √[(3-1)^2 + (6-(-2))^2 + (13-3)^2] PQ = √[4 + 64 + 100] PQ = √168 = 2√42 ∴ Answer: A
Correct Answer: A