Exemplo 2 de Algebra and Trigonometry, 12.1 The Ellipse

Writing the Equation of an Ellipse Centered at a Point Other Than the Origin

Problema e solução em inglês, como no livro original.

What is the standard form equation of the ellipse that has vertices $(−2, −8)$ and $(−2, 2)$

and foci $(−2, −7)$ and $(−2, 1) ?$

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
The x-coordinates of the vertices and foci are the same, so the major axis is parallel to the y-axis. Thus, the equation of the ellipse will have the form

$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$

First, we identify the center, $(h, k) .$ The center is halfway between the vertices, $(- 2, - 8)$ and $(- 2, 2) .$ Applying the midpoint formula, we have:

$\begin{array}{l} (h, k) = (\frac{−2 + (−2)}{2}, \frac{−8 + 2}{2}) \\ = (−2, −3) \end{array}$
Next, we find $a^{2} .$ The length of the major axis, $2 a,$ is bounded by the vertices. We solve for $a$ by finding the distance between the y-coordinates of the vertices.

$\begin{matrix} 2 a = 2 - (−8) \\ 2 a = 10 \\ a = 5 \end{matrix}$
So $a^{2} = 25.$
Now we find $c^{2} .$ The foci are given by $(h, k \pm c) .$ So, $(h, k - c) = (−2, −7)$ and $(h, k + c) = (−2, 1) .$ We substitute $k = −3$ using either of these points to solve for $c .$

$\begin{matrix} k + c = 1 \\ −3 + c = 1 \\ c = 4 \end{matrix}$

So $c^{2} = 16.$

Next, we solve for $b^{2}$ using the equation $\frac{{(x + 2)}^{2}}{9} + \frac{{(y + 3)}^{2}}{25} = 1$

$\begin{matrix} c^{2} = a^{2} - b^{2} \\ 16 = 25 - b^{2} \\ b^{2} = 9 \end{matrix}$
Finally, we substitute the values found for $h, k, a^{2},$ and $b^{2}$ into the standard form equation for an ellipse:

$\frac{{(x + 2)}^{2}}{9} + \frac{{(y + 3)}^{2}}{25} = 1$

Sua vez

O Try It do livro logo depois deste exemplo: mesma ideia, outros números. Só a resposta é dada.

What is the standard form equation of the ellipse that has vertices $(−3, 3)$ and $(5, 3)$ and foci $(1 - 2 \sqrt{3}, 3)$ and $(1 + 2 \sqrt{3}, 3) ?$

Como foi?