The correct option is D 75∘
Let the required line through the point (1,2) be inclined at an angle θ to the axis of x. Then its equation is
x−1cos θ=y−2sin θ=r .....(i)
where r is distance of any point (x, y) on the line from the point (1, 2).
The coordinates of any point on the line (i) are (1+r cos θ,2+r sin θ). If this point is at a distance √63 form (1, 2), then r=√63.
Therefore, the point is (1+√63cos θ,2+√63sin θ) .
But this point lies on the line x+y=4.
⇒√63(cos θ+sin θ)=1orsin θ+cos θ=3√6
⇒1√2sin θ+1√2cos θ=√32,
{Dividing both sides by √2}
⇒sin(θ+45∘)=sin60∘ or sin120∘
⇒θ=15∘ or 75∘.