The maximum distance from origin of a point on the curve x=a Sint−b Sinatb, y=a Cost−b Cosatb, both a, b>0 is

x2+y2=a2Sin2t+b2Sin2atb−2ab Sint.Sinatb+a2Cos2t+b2Cos2atb−2ab CostCosatb=a2+b2−2ab Cost−atb ∴ Max. distance from origin to (x, y) =x2+y2=a2+b2−2ab(−1)=a+b

The maximum distance from origin of a point on the curve x=a Sint−b Sinatb, y=a Cost−b Cosatb, both a, b>0 is

x2+y2=a2Sin2t+b2Sin2atb−2ab Sint.Sinatb+a2Cos2t+b2Cos2atb−2ab CostCosatb=a2+b2−2ab Cost−atb ∴ Max. distance from origin to (x, y) =x2+y2=a2+b2−2ab(−1)=a+b

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The maximum distance from origin of a point on the curve $x = a S int - b S i n (\frac{a t}{b}), y = a C o s t - b C o s (\frac{a t}{b}),$ both $a, b > 0$ is

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$\sqrt{a^{2} - b^{2}}$

$a + b$

$\sqrt{a^{2} + b^{2}}$

$a - b$

answer is D.

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Detailed Solution

$x^{2} + y^{2} = a^{2} S i n^{2} t + b^{2} S i n^{2} (\frac{a t}{b}) - 2 a b Sint . Sin (\frac{a t}{b})$

$\begin{array}{l} + a^{2} C o s^{2} t + b^{2} C o s^{2} (\frac{a t}{b}) - 2 a b C o s t C o s (\frac{a t}{b}) \\ = a^{2} + b^{2} - 2 a b C o s (t - \frac{a t}{b}) \\ ∴ M a x . distance f r o m o r i g i n t o (x, y) \\ = \sqrt{x^{2} + y^{2}} = \sqrt{a^{2} + b^{2} - 2 a b (- 1)} = a + b \end{array}$