The larger of two angles made with the X -axis of a straight line...

Let $A = (1,2)$ equation of line is

$\dfrac{x - 1}{cos\theta} = \dfrac{y - 2}{sin\theta} = r = \dfrac{6}{3}$

$∴ x = \dfrac{6}{3}cos\theta + 1, y = \dfrac{6}{3}sin\theta + 2$

This point lies on line $x + y = 4$

$∴ \dfrac{6}{3}cos\theta + \dfrac{6}{3}sin\theta = 1$

$⇒ sin\theta + cos\theta = \dfrac{3}{2}$

$⇒ \sin \left(\dfrac{\pi}{4} + \theta\right) = \dfrac{3}{2}$

$⇒ \dfrac{\pi}{4} + \theta = 60 °$ or $120 °$

$⇒ \theta = 15 °$ or $75 °$

$∴$ Larger angle $= 75 °$