Let A be the point of intersection of the lines x + y = 3 and 2x - y = 0 . Two variable lines L₁ and L₂ pass through A such that their perpendicular distance from the point P(6, 2) is exactly 4 . If L₁ and L₂ intersect the y-axis at points Q and R respectively, then the length of the line segment QR is

Step-by-step solution

Solving the equations x + y = 3 and 2x - y = 0 , we get 3x = 3 x = 1 . Substituting x = 1 in the first equation gives y = 2 . Thus, the point of intersection is A(1, 2) . Let the equation of a line passing through A(1, 2) with slope m be y - 2 = m(x - 1) , which can be rewritten as mx - y + (2 - m) = 0 . The perpendicular distance from P(6, 2) to this line is given as 4 . Therefore, |m(6) - 2 + (2 - m)| m^2 + 1 = 4 |5m| m^2 + 1 = 4 Squaring both sides, we get: 25m^2 = 16(m^2 + 1) 9m^2 = 16 m^2 = 16 9 m = 4 3 The y-