Math Problem Statement

Trong không gian Oxyz, cho đường thẳng d: x/2 = (y-1)/1 = (z+1)/1 và mặt phẳng (α) : x - 2y - 2z + 5 = 0. Điểm A(a; b; c) có hoành độ dương thuộc đường thẳng d sao cho khoảng cách từ A đến (α) bằng 3. Tính tổng a + b - c?

Solution

The image you've uploaded contains a question in Vietnamese about geometry in a three-dimensional space. Here's a translation of the problem:


Question 1:
In the space OxyzOxyz, given the line dd defined by the system: x2=y11=z+11\frac{x}{2} = \frac{y-1}{1} = \frac{z+1}{1} and the plane (α)(\alpha) defined by the equation: x2y2z+5=0x - 2y - 2z + 5 = 0 Find the coordinates of the point A(a;b;c)A(a; b; c) such that the point lies on the line dd and the distance from AA to the plane α\alpha is 3. Calculate a+bca + b - c.


Approach:

  1. Parameterize the Line dd:
    The equation for the line dd can be written as: x=2t,y=1+t,z=1+tx = 2t, \quad y = 1 + t, \quad z = -1 + t where tt is the parameter.

  2. Substitute Coordinates of A(a;b;c)A(a; b; c) into the Plane Equation:
    The point A(a;b;c)A(a; b; c) lies on the line, so substitute the parameterized coordinates into the plane equation: 2t2(1+t)2(1+t)+5=02t - 2(1 + t) - 2(-1 + t) + 5 = 0

  3. Find the Distance from Point AA to Plane α\alpha:
    The distance dd from a point (x1,y1,z1)(x_1, y_1, z_1) to the plane Ax+By+Cz+D=0Ax + By + Cz + D = 0 is given by the formula: d=Ax1+By1+Cz1+DA2+B2+C2d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} For this problem, we set d=3d = 3 and solve for the coordinates a,b,ca, b, c.

  4. Solve for a+bca + b - c:
    Once the values for aa, bb, and cc are found, compute a+bca + b - c.


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Math Problem Analysis

Mathematical Concepts

Analytical Geometry
Distance from Point to Plane
Parameterization of a Line

Formulas

Distance from point to plane: d = |Ax1 + By1 + Cz1 + D| / sqrt(A^2 + B^2 + C^2)
Parameterization of a line

Theorems

Distance from a point to a plane formula

Suitable Grade Level

Grades 10-12