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Question

 The curve represented by, $${x}=2 (\cos {t}+\sin$$$${t})$$ and $${y}=5(\cos{t} -\sin {t})$$ is


A
Circle
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B
Parabola
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C
Ellipse
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D
Hyperbola
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Solution

The correct option is B Ellipse
Given, $$x=2\left(\cos t+\sin t\right)$$

$$\Rightarrow \dfrac{x}{2}=\cos t+\sin t$$ ...(i)

And $$y=5\left(\cos t-\sin t\right)$$

$$\Rightarrow \dfrac{y}{5}=\cos t-\sin t$$ ...(ii)

Then $$\left(\dfrac{x}{2}\right)^{2}+\left(\dfrac{y}{5}\right)^{2}$$

$$=\left(\cos t+\sin t\right)^{2}+\left(\cos t-\sin t\right)^{2}$$

$$=\cos ^{2}t+\sin ^{2}t+2\cos t.\sin t+\cos ^{2}t+\sin ^{2}t-2\cos t.\sin t$$

$$=2\left(\cos ^{2}t+\sin ^{2}t\right)$$

$$=2$$

Hence, $$\left(\dfrac{x}{2}\right)^{2}+\left(\dfrac{y}{5}\right)^{2}=2$$

$$\dfrac{x^{2}}{4}+\dfrac{y^{2}}{25}=2$$

$$\dfrac{x^{2}}{8}+\dfrac{y^{2}}{50}=1$$

This represents an ellipse.

Maths

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