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1

AIEEE 2002

The maximum distance from origin of a point on the curve
$$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$
$$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$$ both $$a,b > 0$$ is
A
$$a-b$$
B
$$a+b$$
C
$$\sqrt {{a^2} + {b^2}}$$
D
$$\sqrt {{a^2} - {b^2}}$$

Explanation

Distance of origin from $$\left( {x,y} \right) = \sqrt {{x^2} + {y^2}}$$

$$= \sqrt {{a^2} + {b^2} - 2ab\cos \left( {t - {{at} \over b}} \right)} ;$$

$$\le \sqrt {{a^2} + {b^2} + 2ab}$$ $$\left[ {{{\left\{ {\cos \left( {t - {{at} \over b}} \right)} \right\}}_{\min }} = - 1} \right]$$

$$=a+b$$

$$\therefore$$ Maximum distance from origin $$=a+b$$

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