Angles made with the x-axis by two lines drawn through the point (1, 2) and cutting the line x + y = 4 at a distance 63 from the point (1, 2) are

Suppose a line passing through (1, 2) makes an angle θ with x-axis. Then, its equation is          x−1cos⁡θ=y−2sin⁡θThe coordinates of a point on this line at a distance 63=23 from (1, 2) are given by x−1cos⁡θ=y−2sin⁡θ=23 or, 1+23cos⁡θ,2+23sin⁡θThis point lies on x+y=4.∴ 1+23cos⁡θ+2+23sin⁡θ=4⇒ cos⁡θ+sin⁡θ=32⇒ sin⁡θ+π4=32⇒θ+π4=π3,2π3⇒θ=π12 and 5π12