Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202523 Jan 2025Evening ShiftMathematicsStraight LinesActual

A rod of length eight units moves such that its ends A and B always lie on the lines x-y+2=0 and y+2=0 , respectively. If the locus of the point P , that divides the rod A B internally in the ratio 2: 1 is 9 (x^2+ y^2+ x y+ x+28 y )-76=0 , then - - is equal to :

Options

  1. A22
  2. B21
  3. C23
  4. D24

Correct answer

C. 23

Step-by-step solution

aligned & A B=8 & A B^2=64 aligned (a-b)^2+(b+4)^2=64 ...(1) Now P divides A B in the ratio 2: 1 internally h= 2 a+b 3 and k= -4+b+2 3 2 a+b=3 h (2) k= b-2 3 From equation (2) and (3) b=3 k +2 aligned & 2 a=3 h-3 k-2 & a= 3 h-3 k-2 2 aligned Now by putting value of a and b in equation aligned & ( 3 h-3 k-2 2 -(3 k+2) )^2+(3 k+2+4)^2=64 & ( 3 h-3 k-2-6 k-4 2 )^2+(3 k+6)^2=64 & (3 h-9 k-6)^2+4(3 k+6)^2=4 64 & 9(h-3 k-2)^2+36(k+2)^2=256 & 9 (h^2+9 k^2+4-6 h k-4 h+12 k ) & +36 (k^2+4+4 k )=256 & 9 (h^2+13 k^2+20-6 h k-

Practice Straight Lines on Quantrex Academy →

More from Straight Lines

If a straight line drawn through the point of intersection of the lines 4x+3y-1=0 and 3x+4y-1=0 , meets the co-ordinate axes at the points P and Q, then the locus of the mid point 2026From the point (-1, -1) , two rays are sent making angles of 45° with the line x + y = 0 . These rays get reflected from the mirror x + 2y = 1 . If the equations of the reflected r 2026In an equilateral triangle PQR , let the vertex P be at (3, 5) and the side QR be along the line x + y = 4 . If the orthocentre of the triangle PQR is ( , ) , then 9( + ) is equal 2026Let P(3 , 2 ) , 0 , be a point on the ellipse x^2 9 + y^2 4 =1 , Q be a point on the circle x^2+y^2-14x-14y+82=0 and R be a point on the line x+y=5 such that the centroid of the tr 2026Let A,B be points on the two half-lines x- 3 |y|= , >0 at a distance of from their point of intersection P . The line segment AB meets the angle bisector of the given half-lines at 2026Let the line L₁ : x + 3 = 0 intersect the lines L₂ : x - y = 0 and L₃ : 3x + y = 0 at the points A and B , respectively. Let the bisector of the obtuse angle between the lines L₂ a 2026Let the vertex A of a triangle ABC be (1, 2) , and the mid-point of the side AB be (5, -1) . If the centroid of this triangle is (3, 4) and its circumcenter is ( , ) , then 21( + ) 2026Let the mid points of the sides of a triangle ABC be ( 5 2 , 7 ) , ( 5 2 , 3 ) and (4, 5) . If its incentre is (h, k) , then 3h + k is equal to : 2026 Full Straight Lines list All JEE Main PYQs