Step 1: Find the equation of line
is the line of intersection of the planes given by: To find the line of intersection, subtract equation (2) from equation (1): Substitute into equation (2): Now express and in terms of : -
- Let’s use as the parameter. Then: -
-
- To find a point on , set :
- , ,
So, a point on is .
The direction vector of can be found by observing the coefficients of :
- As changes, , , , so the direction vector is .
Thus, the parametric equation of is: Or in symmetric form: Step 2: Find the equation of line
passes through the point and is parallel to . Since is parallel to , it has the same direction vector, . The parametric equation of passing through with direction vector is: Step 3: Find point , the intersection of with plane
Plane is given by: Substitute the parametric equations of into the equation of plane : Now, find the coordinates of by substituting into the equation of :
-
-
- So, point is .
Step 4: Find point , the foot of the perpendicular from to plane
The normal vector to plane , , is . The line from perpendicular to plane has direction vector . The parametric equation of the line from in the direction of the normal is: Find where this line intersects plane : Substitute :
-
-
- So, point is .
Step 5: Evaluate each option (A) The length of the line segment is
- ,
- Vector
- Length of
Option (A) is true. (B) The length of the line segment is 15
- ,
- Vector
- Length of Since , which is not exactly 15, let’s compute more precisely:
- , , so is between 15 and 16, closer to 15 but not exactly 15.
Option (B) is false. (C) The area of is
- Vectors , .
- Compute the cross product : - -component:
- -component:
- -component:
- So, . - Magnitude: . - Area of . Option (C) is true. (D) The acute angle between the line segments and is
- , .
- Dot product: .
- Magnitudes: , .
- .
The given angle has . Compare:
-
-
These values are not equal, so the angles are different. Option (D) is false.
Final Answer: The true statements are:
- (A) The length of the line segment is .
- (C) The area of is .
Thus, the correct options are (A) and (C).