If x = a cos t , y = a sin t , then d x 2 d 2 y at t = 4 π is

x 2 + y 2 = a 2 . Differentiating: 2 x + 2 y y 1 = 0 ⇒ y 1 = − y x , so y 1 ( 4 π ) = − 1 . Differentiating again: 1 + y 1 2 + y y 2 = 0 ⇒ y 2 ( 4 π ) = − y 1 + ( y 1 ) 2 = − a / 2 2 = − a 2 2 .

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MathematicsDifferentiationParametric/Determinant/Logarithmic InverseMedium2 minQB

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If $x = a cos t, y = a sin t$ , then $\frac{d ^{2} y}{d x ^{2}}$ at $t = \frac{π}{4}$ is

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Solution:

$x^{2} + y^{2} = a^{2}$ . Differentiating: $2 x + 2 y y_{1} = 0 \Rightarrow y_{1} = - \frac{x}{y}$ , so $y_{1} (\frac{π}{4}) = - 1$ . Differentiating again: $1 + y_{1}^{2} + y y_{2} = 0 \Rightarrow y_{2} (\frac{π}{4}) = - \frac{1 + ( y _{1} ) ^{2}}{y} = - \frac{2}{a / 2} = - \frac{2 2}{a}$ .

Stream:JEESubject:MathematicsTopic:DifferentiationSubtopic:Parametric/Determinant/Logarithmic Inverse

⏱ 2mℹ️ Source: QB

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