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Math  /  Geometry

QuestionConvert these Cartesian equation to parametric equations in terms tt. \begin{tabular}{|l} (x7)2+(y6)2=1⟹{x(t)=□cos⁡(t)+□y(t)=□sin⁡(t)+□\left(\frac{x}{7}\right)^{2}+\left(\frac{y}{6}\right)^{2}=1 \Longrightarrow\left\{\begin{array}{l}x(t)=\square \cos (t)+\square \\ y(t)\end{array}=\square \sin (t)+\square\right. \end{tabular}

Studdy Solution

STEP 1

1. The given equation represents an ellipse in Cartesian coordinates.
2. The standard form of an ellipse is (xa)2+(yb)2=1\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1.
3. The parametric equations for an ellipse are x(t)=acos⁡(t)+hx(t) = a \cos(t) + h and y(t)=bsin⁡(t)+ky(t) = b \sin(t) + k, where (h,k)(h, k) is the center of the ellipse.

STEP 2

1. Identify the values of aa and bb from the given equation.
2. Write the parametric equations using the identified values.
3. Verify the parametric equations.

STEP 3

Identify the values of aa and bb from the given equation:
Given equation: (x7)2+(y6)2=1\left(\frac{x}{7}\right)^2 + \left(\frac{y}{6}\right)^2 = 1
From this, we identify: a=7 a = 7 b=6 b = 6

STEP 4

Write the parametric equations using the identified values:
The parametric equations for the ellipse are: x(t)=acos⁡(t)+h x(t) = a \cos(t) + h y(t)=bsin⁡(t)+k y(t) = b \sin(t) + k
Since the ellipse is centered at the origin (0,0)(0, 0), h=0h = 0 and k=0k = 0.
Thus, the parametric equations become: x(t)=7cos⁡(t) x(t) = 7 \cos(t) y(t)=6sin⁡(t) y(t) = 6 \sin(t)

STEP 5

Verify the parametric equations:
Substitute x(t)=7cos⁡(t)x(t) = 7 \cos(t) and y(t)=6sin⁡(t)y(t) = 6 \sin(t) back into the original Cartesian equation:
(7cos⁡(t)7)2+(6sin⁡(t)6)2=cos⁡2(t)+sin⁡2(t)=1\left(\frac{7 \cos(t)}{7}\right)^2 + \left(\frac{6 \sin(t)}{6}\right)^2 = \cos^2(t) + \sin^2(t) = 1
The identity cos⁡2(t)+sin⁡2(t)=1\cos^2(t) + \sin^2(t) = 1 is satisfied, confirming the correctness of the parametric equations.
The parametric equations are:
x(t)=7cos⁡(t) x(t) = 7 \cos(t) y(t)=6sin⁡(t) y(t) = 6 \sin(t)

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