Three Dimensional Geometry · Question 1

Let L be the straight line joining the points P(1,2,-1) and Q(2,3,1). Let S be the foot of the perpendicular drawn from the point R(4,-1,5) to the line L. Another line passing through R intersects L at a point T such that the point S divides the line segment PT internally in the ratio |PS|:|ST|=1:2, where |PS| and |ST| are the lengths of the line segments PS and ST, respectively. Then which of the following statements is (are) TRUE ?

1. The orthocentre of the triangle PRT is ( 235,-4, 315 )
2. The orthocentre of the triangle PRT is (4,3,5)
3. The area of the triangle PRT is 6 5
4. The area of the triangle PRT is 18 5

Let L1 be the line of intersection of the planes given by the equations 2x + 3y + z = 4 and x + 2y + z = 5. Let L2 be the line passing through the point P(2, -1, 3) and parallel to L1. Let M denote the plane given by the equation 2x + y - 2z = 6. Suppose that the line L2 meets the plane M at the point Q. Let R be the foot of the perpendicular drawn from P to the plane M. Then which of the following statements is (are) TRUE?

1. The length of the line segment PQ is 9 3
2. The length of the line segment QR is 15
3. The area of PQR is 32 234
4. The acute angle between the line segments PQ and PR is -1 ( 12 3 )

Let R3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7). Let dist(X,Y) denote the distance between two points X and Y in R3. Let S= \X R3 : (dist(X,P) )2- (dist(X,Q) )2=50 \ and T= \Y R3 : (dist(Y,Q) )2- (dist(Y,P) )2=50 \. Then which of the following statements is (are) TRUE?

1. There is a triangle whose area is 1 and all of whose vertices are from S.
2. There are two distinct points L and M in T such that each point on the line segment LM is also in T.
3. There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.
4. There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

Let R be such that the lines L1: x+111= y+212= z+293 and L2: x+163= y+112= z+4 intersect. Let R1 be the point of intersection of L1 and L2. Let O=(0,0,0), and n denote a unit normal vector to the plane containing both the lines L1 and L2. Match each entry in List-I to the correct entry in List-II.

1. (P) → (3) (Q) → (4) (R) → (1) (S) → (2)
2. (P) → (5) (Q) → (4) (R) → (1) (S) → (2)
3. (P) → (3) (Q) → (4) (R) → (1) (S) → (5)
4. (P) → (3) (Q) → (1) (R) → (4) (S) → (5)

A straight line drawn from the point P(1,3,2), parallel to the line x-21= y-42= z-61, intersects the plane L1: x-y+3z=6 at the point Q. Another straight line which passes through Q and is perpendicular to the plane L1 intersects the plane L2: 2x-y+z=-4 at the point R. Then which of the following statements is (are) TRUE?

1. The length of the line segment PQ is 6
2. The coordinates of R are (1,6,3)
3. The centroid of the triangle PQR is ( 43, 143, 53 )
4. The perimeter of the triangle PQR is 2+ 6+ 11