line through the point A (2, 4) intersects the line x + y = 9 at the point P. The minimum distance of AP, is

The equation of a line passing through A(2, 4) is x−2cos⁡θ=y−4sin⁡θSuppose it cuts the line x + y = 9 at point P whose coordinates are given by  x−2cos⁡θ=y−4sin⁡θ=r i.e. x=2+rcos⁡θ,y=4+rsin⁡θ∴    2+rcos⁡θ+4+rsin⁡θ=9⇒    r(cos⁡θ+sin⁡θ)=3⇒    r=3cos⁡θ+sin⁡θ⇒ r≥32 ∵cos⁡θ+sin⁡θ≤2∴1cos⁡θ+sin⁡θ≥12Hence, the minimum value of AP is 32