JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Straight Lines

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{π}{4}

about the origin in the anti-clockwise direction.

If the co-ordinates of the final position of the point P are

(- \frac{1}{2}, \frac{7}{2})

, then the value of

2 a + b

is equal to :

Answer: (D)

Solution

Reflection in the line

y = x

sends

(a, b)

(b, a)

, and the translation then gives

(b + 2, a)

. A rotation through

\frac{π}{4}

about the origin in the anti-clockwise sense carries

(X, Y)

(X cos \frac{π}{4} - Y sin \frac{π}{4}, X sin \frac{π}{4} + Y cos \frac{π}{4})

, so the final position is

(\frac{( b + 2 ) - a}{2}, \frac{( b + 2 ) + a}{2})

Comparing with

(- \frac{1}{2}, \frac{7}{2})

b + 2 - a = - 1, b + 2 + a = 7

Adding the two gives

2 (b + 2) = 6

, so

b = 1

, and then

a = 4

. Hence

2 a + b = 9

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Single correct option