Let ( x , y ) be a point lying on 2 x α − 1 2 y 2 β 2 6 3 γ 1 + 2 x 4 γ 2 y 2 α − 1 1 3 β 2 + 11 72 − 2 γ 2 α − 81 β = 0 ∀ α , β , γ ∈ R If the minimum distance between ( x

Question

Let ( x , y ) be a point lying on ​ 2 x α − 1 ​ 2 y 2 β 2 ​ 6 3 γ 1 ​ ​ + 2 ​ x 4 γ 2 ​ y 2 α − 1 ​ 1 3 β 2 ​ ​ + ​ 11 72 ​ − 2 γ 2 α ​ ​ − 81 β = 0 ∀ α , β , γ ∈ R If the minimum distance between ( x , y ) and the line 4 x − 3 y + 12 = 0 is d , then the value of d is

TestHub · Accepted Answer

2 x − ( 2 − 6 γ ) − 2 y ( α + 3 γ ) + 6 ( 2 α + 2 β ) + 2 x ( 4 α + 3 β ) − 2 y ( 8 γ − 6 β ) + 2 ( − 4 γ − 4 α ) + 22 α + 144 γ − 81 β = 0 ∴ α ( 8 x − 2 y + 26 ) + β ( 10 x − 12 y − 69 ) + γ ( − 12 x − 22 y + 136 ) = 0 4 x − y + 13 = 0 10 x + 12 y − 69 = 0 6 x + 11 y − 68 = 0 ​ ⎭ ⎬ ⎫ ​ x = − 2 3 ​ , y = 7 Therefore, f ( x , y ) is at point ( − 2 3 ​ , 7 ) . d = ​ 5 4 ( − 2 3 ​ ) − 3 ( 7 ) + 12 ​ ​ = 3

Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

Doubts & Discussion

JEE Advanced (2028)