Graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 1

검증된 단계별 안내

Identify the standard form of the ellipse equation: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.

From the given equation \(\frac{(x - 2)^2}{7} + \frac{(y + 1)^2}{11} = 1\), determine the center of the ellipse as \((2, -1)\).

Compare the denominators to find \(a^2\) and \(b^2\). Since \(11 > 7\), set \(a^2 = 11\) and \(b^2 = 7\). This means the major axis is vertical.

Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Substitute the values to find \(c = \sqrt{11 - 7}\).

Locate the foci along the major axis (vertical axis) by adding and subtracting \(c\) from the \(y\)-coordinate of the center. The foci are at \((2, -1 + c)\) and \((2, -1 - c)\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Standard Form of an Ellipse

An ellipse can be expressed in the standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center. The denominators \(a^2\) and \(b^2\) represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Identifying these values helps in graphing the ellipse accurately.

Determining the Orientation of the Ellipse

The larger denominator between \(a^2\) and \(b^2\) indicates the major axis direction: if \(a^2 > b^2\), the ellipse is stretched horizontally; if \(b^2 > a^2\), it is stretched vertically. This orientation is crucial for locating the foci and sketching the ellipse.

Finding the Foci of an Ellipse

The foci lie along the major axis, located at a distance \(c\) from the center, where \(c = \sqrt{|a^2 - b^2|}\). Knowing \(c\) and the center coordinates allows you to find the exact positions of the foci, which are key points defining the ellipse's shape.