The number of tangents to the curve y = cos ( x + y ) , ∣ x ∣ ≤ 2 π , that are parallel to the line x + 2 y = 0 is

y ′ = − sin ( x + y ) ( 1 + y ′ ) Slope of tangent is − 2 1 = y ′ ⇒ 2 1 = 2 s i n ( x + y ) ⋅ 2 ⇒ sin ( x + y ) = 1 , cos ( x + y ) = 0 ⇒ y = 0 ⇒ 0 = cos x ⇒ x = 2 π , 2 − 3 π (both in [ − 2 π , 2 π ] ), which satisfy the above equation. ∴ 2 tangents

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MathematicsApplication of DerivativeTangents & NormalsMedium2 minQB

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The number of tangents to the curve $y = cos (x + y), ∣ x ∣ \leq 2 π$ , that are parallel to the line $x + 2 y = 0$ is

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$y^{'} = - sin (x + y) (1 + y^{'})$ Slope of tangent is $- \frac{1}{2} = y^{'} \Rightarrow \frac{1}{2} = \frac{s i n ( x + y )}{2} \cdot 2 \Rightarrow sin (x + y) = 1, cos (x + y) = 0$ $\Rightarrow y = 0 \Rightarrow 0 = cos x \Rightarrow x = \frac{π}{2}, \frac{- 3 π}{2}$ (both in $[- 2 π, 2 π]$ ), which satisfy the above equation. $∴ 2$ tangents

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Tangents & Normals

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