Question

A point P(x, y) moves in xy plane in such a way that $\sqrt { 2 }$ ≤ |x + y| + |x + y| ≤ 2 $\sqrt { 2 }$ . Area of the region representing all possible positions of the point 'P', is equal to

• A. 2 sq. units
• B. 4 sq. units
• C. 6 sq. units
• D. 8 sq. union

Answer 1

Answer: (C)

6 sq. units

Answer 2

For y – x ≤ 0, x + y ≥ 0.

We get, $\sqrt { 2 }$ ≤ x + y + x – y ≤ $2 \sqrt { 2 }$

⇒ $\frac { 1 } { \sqrt { 2 } } \leq x \leq \sqrt { 2 }$

for y − x ≥ 0, x + y ≥ 0 We get, $\frac { 1 } { \sqrt { 2 } } \leq y \leq \sqrt { 2 }$

For x + y ≤ 0, y - x ≥ 0 We get, $- \sqrt { 2 } \leq x \leq - \frac { 1 } { \sqrt { 2 } }$

For x + y ≤ 0, y - x ≤ 0 We get, $- \sqrt { 2 } \leq y \leq - \frac { 1 } { \sqrt { 2 } }$

Shaded region represents all positions of point P. it's area is equal to ∆_ABCD − ∆_A1B1C1D1

i.e. $( 2 \sqrt { 2 } ) ^ { 2 } - ( \sqrt { 2 } ) ^ { 2 }$ i.e. 6 sq. units.