The point P (a, b) undergoes the following three transformations successively : (a) reflec…

Question

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 $${\pi  \over 4}$$ about the origin in the anti-clockwise direction.

If the co-ordinates of the final position of the point P are $$\left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$$, then the value of 2a + b is equal to :

Accepted Answer

Image of A(a, b) along y = x is B(b, a). Translating it 2 units it becomes C(b + 2, a).

Now, applying rotation theorem

$$ - {1 \over {\sqrt 2 }} + {7 \over {\sqrt 2 }}i = \left( {(b + 2) + ai} \right)\left( {\cos {\pi  \over 4} + i\sin {\pi  \over 4}} \right)$$

$$ - {1 \over {\sqrt 2 }} + {7 \over {\sqrt 2 }}i = \left( {{{b + 2} \over {\sqrt 2 }} - {a \over {\sqrt 2 }}} \right) + i\left( {{{b + 2} \over {\sqrt 2 }} + {a \over {\sqrt 2 }}} \right)$$

$$\Rightarrow$$ b - a + 2 = -1 ......(i)

and b + 2 + a = 7 ...... (ii)

$$\Rightarrow$$ a = 4; b = 1

$$\Rightarrow$$ 2a + b = 9

Solutions · 1