QuestionDetermine the general equation and complete solution for the ellipse: $\frac{x^{2}}{4}+\frac{(y+3)^{2}}{9}=1$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is of an ellipse. . The general form of an ellipse equation is $\frac{(x-h)^{}}{a^{}}+\frac{(y-k)^{}}{b^{}}=1$ where $(h,k)$ is the center of the ellipse, $a$ is the semi-major axis, and $b$ is the semi-minor axis.
3. The given equation is already in the general form.

STEP 2

Identify the center $(h,k)$ of the ellipse by comparing the given equation with the general form.
From the given equation $\frac{x^{2}}{4}+\frac{(y+)^{2}}{9}=1$ , we can see that $h=0$ and $k=-$ .

STEP 3

Identify the lengths of the semi-major axis $a$ and the semi-minor axis $b$ by comparing the given equation with the general form.
From the given equation $\frac{x^{2}}{}+\frac{(y+3)^{2}}{9}=1$ , we can see that $a^{2}=$ and $b^{2}=9$ .

STEP 4

Calculate the lengths of the semi-major axis $a$ and the semi-minor axis $b$ .
$a = \sqrt{a^{2}} = \sqrt{4} =2$ $b = \sqrt{b^{2}} = \sqrt{9} =3$

STEP 5

The general equation of the ellipse is $\frac{(x-0)^{2}}{2^{2}}+\frac{(y+3)^{2}}{3^{2}}=1$ which simplifies to $\frac{x^{2}}{4}+\frac{(y+3)^{2}}{9}=1$ .
The complete solution for the ellipse includes the center $(h,k)=(0,-3)$ , the semi-major axis $a=2$ , and the semi-minor axis $b=3$ .

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