Question

Finding Parametric Equations and Tangent LinesFind parametric equations for the given curve.9x² + 4y² = 36

Accepted Answer

Recognize that the given equation $$9x^{2}$$ + $$4y^{2} = 36$ represents an ellipse. The standard form of an ellipse is $$\frac{$$x^{2}$$}{$$a^{2}$$} + \frac{$$y^{2}$$}{$$b^{2}$$} = 1$$. To rewrite the equation in this form, divide both sides by 36 to get $$\frac{$$9x^{2}$$}{36} + \frac{$$4y^{2}$$}{36} = 1$$, which simplifies to $$\frac{$$x^{2}$$}{4} + \frac{$$y^{2}$$}{9} = 1$$. Identify the values of a$ and b$ from the standard form: here, a$$^{2}$$ = 4 so$$ $$a = 2$$, and $$b^{2} = 9$ so b = 3$. This means the ellipse is centered at the origin with semi-major axis length 3 along the y$-axis and semi-minor axis length 2 along the x$-axis. Use the standard parametric equations for an ellipse centered at the origin: x = a$$ \cos(t)$$ and y = b$$ \sin(t)$$, where t$ is the parameter (usually representing an angle). Substitute the values of a$ and b$ to get x = 2$$ \cos(t)$$ and y = 3$$ \sin(t)$$. Write down the parametric equations explicitly:

$$
\begin{cases}
 x(t) = 2 \cos(t) \
 y(t) = 3 \sin(t)
\end{cases}
$$

where t$ varies over an interval such as $$[0, 2\pi]$$ to trace the entire ellipse. To find the tangent line at a specific parameter value t$$ = $$t_0$$, first compute the derivatives $$\frac{dx}{dt} = -2 \sin(t)$$ and $$\frac{dy}{dt} = 3 \cos(t). $$Then, the slope of the tangent line at $$t_0 is $$given by $$\frac{dy/dt}{dx/dt} = \frac{3 \cos(t_0$)}{-2 \sin(t_0$)}. $$Use the point $$(x(t_0$), y(t_0$))$$ and this slope to write the equation of the tangent line.

Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

9x² + 4y² = 36

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Key Concepts

Parametric Equations

Ellipse Equation and Standard Form

Tangent Lines to Parametric Curves