The point (4$$, I $$) undergoes the following three transformations successively.i . Reflection about the line y: x.ii. Translation through a distance $$2$$ units along the positive direction of the $$x-axis$$. iii. Rotation through an angle π$$4$$ about the origin in the counterclockwise direction. Then the final position of the point is given by the co ordinates

Question

The point (4$$, I $$) undergoes the following three transformations successively.i . Reflection about the line y: x.ii. Translation through a distance $$2$$ units along the positive direction of the $$x-axis$$. iii. Rotation through an angle π$$4$$ about the origin in the counterclockwise direction. Then the final position of the point is given by the co ordinates

Infinity Learn · Accepted Answer

Reflection about the line y $$=$$ x changes the point (4$$, $$l) to (1$$, $$4). On the translation of (1$$, $$4) through a distance of $$2$$ units along the positive direction of the $$x-axis$$, the point becomes (1$$ $$+$$ $$2$$. $$4), i.e., (3$$,$$4). On rotation about the origin through an angle π$$4point$$ P takes the position P' such that $$OP=OP$$′. Also, $$OP=5=OP$$′ and $$cos$$⁡θ$$=3/5$$,$$sin$$⁡θ$$=4/5$$ . Now, $$x=OP$$′$$cos$$⁡π$$4+$$θ$$=5cos$$⁡π$$4cos$$⁡θ−$$sin$$⁡θ$$=5352$$−$$452=$$−$$12y=OP$$′$$sin$$⁡π$$4+$$θ$$=5sin$$⁡π$$4cos$$⁡θ$$+$$$$cos$$⁡π$$4sin$$⁡θ$$=5352+452=72P$$′≡−$$12$$,$$72$$

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