Three Dimensional Geometry · Question 3

JEE Advanced 2025 Paper 1Multiple CorrectEasy📈 Topic in 50% papers+4−1

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?

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

Answer

(1) The length of the line segment PQ is 9 3, (3) The area of PQR is 32 234

Solution

Direction of L1: (2 i + 3 j + k) ( i + 2 j + k) = i - j + k.

L2: (2 + t, -1 - t, 3 + t). In M: 2(2+t) + (-1-t) - 2(3+t) = 6 -3 - t = 6 t = -9, so Q = (-7, 8, -6) and PQ = 9 3. (A) true.

Foot R: P + (2, 1, -2) with 9 = 6 - (4 - 1 - 6) = 9 = 1, R = (4, 0, 1), PR = 3.

PRQ = 90 , so QR = PQ2 - PR2 = 243 - 9 = 234 15. (B) false.

Area = 12 PR QR = 32 234. (C) true.

QPR = PRPQ = 39 3 = 13 3 12 3. (D) false.

Answer: (A), (C)

JEE Advanced 2025 Paper 1

Keep practising — Three Dimensional Geometry

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?

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

JEE Advanced 2024 Paper 1

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.

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

JEE Advanced 2024 Paper 1

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?

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

JEE Advanced 2024 Paper 2

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