Foot Of Perpendicular And Image Of A Point

Question

Given the point (1, 0, 0) and its reflection across the plane x + y + z = 3, identify the image point coordinates.

RankGuru · Accepted Answer

The procedure is the image-of-a-point-in-a-plane formula: for plane ax + by + cz + d = 0 and point P, the image Q satisfies (x' - x_1)/a = (y' - y_1)/b = (z' - z_1)/c = -2(ax_1 + by_1 + cz_1 + d)/(a^2 + b^2 + c^2). This reflection rule is a core JEE Advanced result built on moving twice the perpendicular distance along the normal. Writing the plane as x + y + z - 3 = 0, we have a = b = c = 1, d = -3, and P = (1, 0, 0). Then ax_1 + by_1 + cz_1 + d = 1 + 0 + 0 - 3 = -2, and a^2+b^2+c^2 = 3, so the common ratio is -2(-2)/3 = 4/3. Thus x' = 1 + 4/3 = 7/3 and y' = z' = 0 + 4/3 = 4/3, giving the image (7/3, 4/3, 4/3). Option (5/3, 2/3, 2/3) is the foot of perpendicular, not the image. Option (3, 2, 2) overshoots the normal step. Option (1/3, 4/3, 4/3) flips the sign on x. This applies the planar reflection theorem. Plausibility check: the midpoint of P and its image is (5/3, 2/3, 2/3), which lies on the plane since 5/3 + 2/3 + 2/3 = 3, confirming correct reflection.

Foot Of Perpendicular And Image Of A Point

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