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

Question

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

Infinity Learn · Accepted Answer

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$$+$$π$$4=32$$⇒θ$$+$$π$$4=$$π$$3$$,$$2$$$$π$$$$3$$⇒θ$$=$$π$$12$$ and $$5$$$$π$$$$12$$

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 $\frac{\sqrt{6}}{3}$ from the point $(1, 2)$ are

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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

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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 $\frac{\sqrt{6}}{3}$ from the point $(1, 2)$ are