99 Percentile Qs Bank for JEE MainMathematicsStraight Lines
Let a ≠ 0 ,   b ≠ 0 ,   c be three real numbers and L p , q = a p + b q + c a 2 + b 2 ,   ∀   p ,   q ∈ R . If L 2 3 , 1 3 + L 1 3 , 2 3 + L ( 2 , 2 ) = 0 , then the line a x + b y + c = 0 always passes through the fixed point
Options
- A0 , 1
- B1 , 1
- C2 , 2
- D- 1 , - 1
Correct answer
B. 1 , 1
Step-by-step solution
Given that L p , q = a p + b q + c a 2 + b 2 . L 2 3 , 1 3 = a 2 3 + b 1 3 + c a 2 + b 2 = 2 a + b + 3 c 3 a 2 + b 2 L 1 3 , 2 3 = a 1 3 + b 2 3 + c a 2 + b 2 = a + 2 b + 3 c 3 a 2 + b 2 L 2 , 2 = a 2 + b 2 + c a 2 + b 2 = 2 a + 2 b + c a 2 + b 2 = 6 a + 6 b + 3 c 3 a 2 + b 2 L 2 3 , 1 3 + L 1 3 , 2 3 + L 2 , 2 = 2 a + b + 3 c 3 a 2 + b 2 + a + 2 b + 3 c 3 a 2 + b 2 + 6 a + 6 b + 3 c 3 a 2 + b 2 = 9 a + 9 b + 9 c 3 a 2 + b 2 = 0 ∴   9 a + 9 b + 9 c = 0 ⇒ a + b + c = 0       . . . 1 B