Question

The point (4, 1) undergoes the following three transformations successively: (a) Reflection about the line y = x (b) Translation through a distance 2 units along the positive direction of the x-axis. (c) Rotation through an angle `pi/4` about the origin in the anti clockwise direction. The final position of the point is given by the co-ordinates.

Answer 1

Reflection of P(4,1) about the line `y=x` is Q(1,4). On translation of Q(1,4) through a distance of 2 units strong positive direction of x-axis, the point moves to `R(1+2,4)` or `R(3,4)`.
On rotation about origin through an angle of `pi//4`, the point R takes the position S such that `OR=OS=5`.

From the figure, `cos theta=(3/5) ` and `sintheta= (4/5)` form trigonometric concepts, coordinates of point S are :
`(5cos (pi/(4+theta)),=5((pi)/(4)+theta))`
Now , `5cos ((pi)/(4)+theta)=5("cos"(pi)/(4)costheta-"sin"(pi)/(4)sintheta)`
`=5((1)/(sqrt2).(1)/(5)-(1)/(sqrt2).(4)/(5))=-(1)/(sqrt2)`
and `5 sin ((pi)/(4)+theta)=5("sin"(pi)/(4)costheta+"sin"thetacos"(pi)/(4))`
`((1)/(sqrt(2)).(1)/(5)+(1)/(sqrt(2)).(4)/(5))=(7)/(sqrt(2))`
Coordinates of S are `(-(1)/(sqrt(2)),(7)/(sqrt2))`.