The line joining the points $A(2,0)$ and $B(3,1)$ is rotated through an angle of $45^{\circ}$ about $A$ in the anti-clockwise direction. Find the coordinates of $B$ in the new position.

The point $P(a, b)$ undergoes the following three transformations successively:
$(a)$ reflection about the line $y=x$.
$(b)$ translation through $2$ units along the positive direction of $x$-axis.
$(c)$ rotation through angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
If the coordinates of the final position of the point $P$ are $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$,then the value of $2a+b$ is equal to:

The point $(3,2)$ undergoes the following three transformations in the order given:
$(i)$ Reflection about the line $y=x$.
(ii) Translation by the distance $1$ unit in the positive direction of $x$-axis.
(iii) Rotation by an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
Then,the final position of the point is:

The coordinates of the feet of the perpendiculars from the vertices of a triangle to the opposite sides are $(20, 25)$,$(8, 16)$,and $(8, 9)$. The orthocentre of the triangle lies at the point-

Let $A = (2, 0)$ and $B = (6, 4)$ be two points. If the line segment $\overline{AB}$ is rotated about $A$ through an angle of $45^{\circ}$ in the negative (clockwise) direction,then the coordinates of $B$ after the rotation are:

The point $(4,1)$ undergoes the following transformations successively:
$I$. Reflection about the line $y=x$.
$II$. Translation through a distance $2$ units in the direction of the positive $X$-axis.
$III$. Rotation through an angle $\frac{\pi}{4}$ about the origin in the anticlockwise direction.
Then,the final position of the point is: