Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsThree Dimensional Geometry

Let P(a, 3a, 2a) be a point in space. Let Q and R be the feet of the perpendiculars drawn from P to the lines L₁: x 2 = y 1 = z 2 and L₂: x -1 = y 2 = z 2 respectively. If the area of the triangle PQR is 10 , then the value of a^2 is ________

Correct answer

4

Step-by-step solution

Let the general point on L₁ be Q(2r, r, 2r) . The direction ratios of L₁ are 2, 1, 2 . The vector PQ = (2r - a) i + (r - 3a) j + (2r - 2a) k . Since Q is the foot of the perpendicular, PQ is perpendicular to L₁ : PQ (2 i + j + 2 k ) = 0 2(2r - a) + 1(r - 3a) + 2(2r - 2a) = 0 4r - 2a + r - 3a + 4r - 4a = 0 9r - 9a = 0 r = a Thus, Q (2a, a, 2a) and PQ = a i - 2a j + 0 k . Similarly, let the general point on L₂ be R(-k, 2k, 2k) . The direction ratios of L₂ are -1, 2, 2 . The vector PR = (-k - a) i + (2k - 3a) j + (2k

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let P be the plane such that it contains the straight line x-1 2 = y-3 3 = z+2 1 and is perpendicular to the plane x + 2y + 3z = 4 . Let P₁ be the plane which passes through the po 2026Let 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 pass 2026Let the foot of perpendicular from the point ( , 2, 3) on the line x-4 1 = y-9 2 = z-5 1 be the point (1, , 2) . Then the distance between the lines x-1 2 = y-2 3 = z+4 6 and x- 2 2026Let a line L₁ pass through the origin and be perpendicular to the lines L₂: r = (3+t) i + (2t-1) j + (2t+4) k and L₃: r = (3+2s) i + (3+2s) j + (2+s) k , t, s R . If (a, b, c) , a 2026The shortest distance between the lines x-4 1 = y-3 2 = z-2 -3 and x+2 2 = y-6 4 = z-5 -5 is : 2026Let the image of the point P(0, -5, 0) in the line x-1 2 = y 1 = z+1 -2 be the point R and the image of the point Q (0, -1 2 , 0 ) in the line x-1 -1 = y+9 4 = z+1 1 be the point S 2026Let the image of the point P(1, 6, a) in the line L: x 1 = y-1 2 = z-a+1 b , b > 0 , be ( a 3 , 0, a+c ) . If S( , , ) , > 0 , is the point on L such that the distance of S from th 2026Let a line L be perpendicular to both the lines L₁: x+1 3 = y+3 5 = z+5 7 and L₂: x-2 1 = y-4 4 = z-6 7 . If is the acute angle between the lines L and L₃: x - 8 7 2 = y - 4 7 1 = 2026 Full Three Dimensional Geometry list All JEE Main PYQs