Family of lines represented by the equation ( cos θ+ sin θ) x +( cos θ- sin θ) y -3(3 cos θ+ sin θ)=0 passes through a fixed point M for all real values of θ. The reflection of M in the line x-y=0, is

Question

Family of lines represented by the equation ( cos θ+ sin θ) x +( cos θ- sin θ) y -3(3 cos θ+ sin θ)=0 passes through a fixed point M for all real values of θ. The reflection of M in the line x-y=0, is (A) (6, 3) (B) (3, 6) (C) (-6, 3) (D) (3, -6)

Tardigrade Admin · Accepted Answer

Given equation is cos θ(x+y-9)+ sin θ(x-y-3)=0 ⇒ (x+y-9)+ tan θ(x-y-3)=0, which is of the form L 1+λ L 2=0 where L1: x+y-9=0 and L2: x-y-3=0 Hence the lines will always pass through the point of intersection of L 1=0 and L 2=0 i.e., M (6,3) So its reflection in the line y = x will be (3,6)

Q. Family of lines represented by the equation (cosθ+sin θ)x+(cosθ−sinθ)y−3(3cosθ+sinθ)=0 passes through a fixed point M for all real values of θ. The reflection of M in the line x−y=0, is

(6, 3)

(3, 6)

(-6, 3)

(3, -6)

Q. Family of lines represented by the equation $(cos θ + sin$ $θ) x + (cos θ - sin θ) y - 3 (3 cos θ + sin θ) = 0$ passes through a fixed point $M$ for all real values of $θ$ . The reflection of $M$ in the line $x - y = 0$ , is