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Question: The curve represented by \(x = 3(\cos t + \sin t),y = 4(\cos t - \sin t)\) is...

The curve represented by x=3(cos⁡t+sin⁡t),y=4(cos⁡t−sin⁡t)x = 3(\cos t + \sin t),y = 4(\cos t - \sin t) is

A

Ellipse

B

Parabola

C

Hyperbola

D

Circle

Answer

Ellipse

Explanation

Solution

Given, x=3(cos⁡t+sin⁡t),y=4(cos⁡t−sin⁡t)x = 3(\cos t + \sin t),y = 4(\cos t - \sin t) ⇒

x3=(cos⁡t+sin⁡t),y4=(cos⁡t−sin⁡t)\frac{\mathbf{x}}{\mathbf{3}}\mathbf{= (}\mathbf{\cos}\mathbf{t}\mathbf{+}\mathbf{\sin}\mathbf{t}\mathbf{),}\frac{\mathbf{y}}{\mathbf{4}}\mathbf{= (}\mathbf{\cos}\mathbf{t}\mathbf{-}\mathbf{\sin}\mathbf{t}\mathbf{)} Squaring and adding, we get x29+y216=(1+sin⁡2t)+(1−sin⁡2t)\frac{x^{2}}{9} + \frac{y^{2}}{16} = (1 + \sin 2t) + (1 - \sin 2t) ⇒ x29+y216=2\frac{x^{2}}{9} + \frac{y^{2}}{16} = 2, which

represents ellipse.