Question

Obtain a Cartesian equation of the curve defined parametrically by the solution of the linear system in Example6. Identify the curve passing through (2, -1) in Figure 8.2.5[Hint: Compute x²,y², andx,y.]

Short answer

The Cartesian equation of the curve is$x^2+4xy+8y^2=4$.

Step-by-step solution

Determine the compute matrix

In general, and especially for matrix calculation, we simply count the number of basic operations such as addition, subtraction, multiplication, and division.

Determine the system of equations

Given

$x=2\cos 2t-2\sin 2t,$and $y=-\cos 2t$

we can substitute the equation of y into x, which yields

$x=-2y-2\sin 2t$

$x+2y=-2\sin 2t$

Square both sides of the equation,

$(x+2y{)}^2=(-2\sin 2t{)}^2$

$x^2+4xy+4y^2=4{\sin }^{2}2t$

Using the trigonometric identity

${\sin }^{2}2t=1-{\cos }^{2}2t$

$x^2+4xy+4y^2=4-4{\cos }^{2}2t$

Recall$y=-\cos 2t$, squaring both sides, $y^2={\cos }^{2}2t$, now substituting,

$x^2+4xy+4y^2=4-4y^2$

or,

$x^2+4xy+8y^2=4$

Therefore, the Cartesian equation of the curve is$x^2+4xy+8y^2=4$.