Let and be integers.
If the equation of the tangent line to the curves and can be expressed as , where
Find:
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f ( x ) = − x 2 and g ( x ) = x ⟹
d x d ( f ( x ) ) ∣ x = a = − 2 a and d x d ( g ( x ) ) ∣ x = b = 2 b 1 ⟹
− 2 a = 2 b 1 ⟹ a = − 4 b 1
A : ( a , − a 2 ) = ( − 4 b 1 , − 1 6 b 1 ) B : ( b , b ) ⟹
slope m = 2 b 1 = 4 b 1 ( 4 b 2 3 + 1 1 6 b 2 3 + 1 ) ⟹
1 6 b 2 3 + 1 = 8 b 2 3 + 2 ⟹ 8 b 2 3 = 1 ⟹ b = 4 1
⟹ slope m = 1 and a = − 2 1
Using A : ( − 2 1 , − 4 1 ) ⟹ y + 4 1 = x + 2 1 ⟹
4 x − 4 y + 1 = 0
∴ ∣ a ∣ = 4 , ∣ b ∣ = 4 , ∣ c ∣ = 1 and g c f ( ∣ a ∣ , ∣ b ∣ , ∣ c ∣ ) = 1 ⟹
∣ a ∣ + ∣ b ∣ + ∣ c ∣ = 9 .
4 x − 4 y + 1 = 0 is tangent to g ( x ) at B : ( 4 1 , 2 1 ) and f ( x ) at A : ( − 2 1 , − 4 1 )