Let a = i ^ + j ^ + k ^ , b = 2 i ^ + 2 j ^ + k ^ and c = 5 i ^ + j ^ - k ^ be three vectors. The area of the region formed by the set of points whose position vectors r → satisfy the equations r · a = 5 and | r - b | + | r - c | = 4 is closest to the integer.

Step-by-step solution

Since, | r - b | + | r - c | = 4 is equation of ellipsoid in plane r , a = 5 or x + y + z = 5 . ∴ Distance between foci = | b - c | = | 3 i ^ - j ^ - 2 k ^ | = 14 = 2 a e and 2 a = 4 ∴   e = 14 4 ⇒ e 2 = 1 - b 2 a 2 ⇒ 14 16 = 1 - b 2 4 ⇒ b 2 4 = 2 16 b 2 = 1 2 ∴ Area of ellipsoid = π a b = π 2 1 2 = 2 π And from the given options the closest integer to 2 π is 4 .