Graph each ellipse and give the location of its foci. (x − 2)²/9 + (y -1)² /4= 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 \(\frac{(x - 3)^2}{36} + \frac{(y - 4)^2}{25} = 1\), determine the center as \((3, 4)\).

Compare the denominators to find \(a^2\) and \(b^2\). Since 36 > 25, \(a^2 = 36\) and \(b^2 = 25\). This means the major axis is horizontal because \(a^2\) is under the \((x - h)^2\) term.

Calculate the lengths of the semi-major axis \(a = \sqrt{36}\) and the semi-minor axis \(b = \sqrt{25}\).

Find the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Then, locate the foci at \((h \pm c, k)\) because the major axis is horizontal.

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

The standard form of an ellipse equation is (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center. The values a² and b² represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Understanding this form helps in identifying the ellipse's size, shape, and position on the coordinate plane.

Major and Minor Axes

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. The lengths of these axes are 2a and 2b, where a and b are the square roots of the denominators in the ellipse equation. Knowing which axis is major or minor depends on whether a² > b² or vice versa.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations are found using c² = a² - b², where c is the distance from the center to each focus along the major axis. Identifying the foci is essential for graphing and understanding ellipse properties.