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Trigonometric Parameters, (Simultaneous Equation?)

The question:
Find the cartesian equation of this parametric form
x=sin(a)+2cos(a)[br]y=2sin(a)+cos(a)x = sin(a) + 2cos(a)[br]y = 2 sin(a) + cos(a)

What I've tried:
Attempt 1
x2=4cos2(a)+3sin(a)cos(a)+sin2(a)[br]y2=4sin2(a)+3sin(a)cos(a)+cos2(a)x^2 = 4cos^2(a) + 3sin(a)cos(a) + sin^2(a)[br]y^2 = 4sin^2(a) + 3sin(a)cos(a) + cos^2(a)

y2+cos2(a)=x2+3sin2(a)[br]y2+3=x2+6sin2(a)y^2 + cos^2(a) = x^2 + 3sin^2(a)[br]y^2 + 3 = x^2 + 6sin^2(a)
Attempt 2
x+sin(a)=y+cos(a)[br]y−x=sin(a)−cos(a)x + sin(a) = y + cos(a)[br]y-x = sin(a) - cos(a)

So both attempts have been fairly directionless and I've had a look at the answer: 5x2+5y2−8xy=95x^2 + 5y^2 -8xy = 9 yet I have no idea where it could have got that from.

If anyone could point me in the right direction it would be much appreciated.
Original post by Retsek
The question:
Find the cartesian equation of this parametric form
x=sin(a)+2cos(a)[br]y=2sin(a)+cos(a)x = sin(a) + 2cos(a)[br]y = 2 sin(a) + cos(a)

What I've tried:
Attempt 1
x2=4cos2(a)+3sin(a)cos(a)+sin2(a)[br]y2=4sin2(a)+3sin(a)cos(a)+cos2(a)x^2 = 4cos^2(a) + 3sin(a)cos(a) + sin^2(a)[br]y^2 = 4sin^2(a) + 3sin(a)cos(a) + cos^2(a)

y2+cos2(a)=x2+3sin2(a)[br]y2+3=x2+6sin2(a)y^2 + cos^2(a) = x^2 + 3sin^2(a)[br]y^2 + 3 = x^2 + 6sin^2(a)
Attempt 2
x+sin(a)=y+cos(a)[br]y−x=sin(a)−cos(a)x + sin(a) = y + cos(a)[br]y-x = sin(a) - cos(a)

So both attempts have been fairly directionless and I've had a look at the answer: 5x2+5y2−8xy=95x^2 + 5y^2 -8xy = 9 yet I have no idea where it could have got that from.

If anyone could point me in the right direction it would be much appreciated.


Standard trig. identity that springs to mind is cos⁡2a+sin⁡2a=1\cos^2a+\sin^2a=1

So, I'd eliminate sin a, from the two equations, to get cos a in terms of x and y.

Similarly cos a.

Then sub into the identity.
Original post by Retsek
The question:
Find the cartesian equation of this parametric form
x=sin(a)+2cos(a)[br]y=2sin(a)+cos(a)x = sin(a) + 2cos(a)[br]y = 2 sin(a) + cos(a)

What I've tried:
Attempt 1
x2=4cos2(a)+3sin(a)cos(a)+sin2(a)[br]y2=4sin2(a)+3sin(a)cos(a)+cos2(a)x^2 = 4cos^2(a) + 3sin(a)cos(a) + sin^2(a)[br]y^2 = 4sin^2(a) + 3sin(a)cos(a) + cos^2(a)

y2+cos2(a)=x2+3sin2(a)[br]y2+3=x2+6sin2(a)y^2 + cos^2(a) = x^2 + 3sin^2(a)[br]y^2 + 3 = x^2 + 6sin^2(a)
Attempt 2
x+sin(a)=y+cos(a)[br]y−x=sin(a)−cos(a)x + sin(a) = y + cos(a)[br]y-x = sin(a) - cos(a)

So both attempts have been fairly directionless and I've had a look at the answer: 5x2+5y2−8xy=95x^2 + 5y^2 -8xy = 9 yet I have no idea where it could have got that from.

If anyone could point me in the right direction it would be much appreciated.


you should have 4sinacosa in your x2 expansion

and also in your y2

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