Graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36

검증된 단계별 안내

Rewrite the given equation in standard form by dividing both sides by 36: \(\frac{9(x - 1)^2}{36} + \frac{4(y + 3)^2}{36} = \frac{36}{36}\).

Simplify the fractions to get: \(\frac{(x - 1)^2}{4} + \frac{(y + 3)^2}{9} = 1\).

Identify the values of \(a^2\) and \(b^2\) from the denominators. Here, \(a^2 = 9\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is vertical.

Find the center of the ellipse, which is at the point \((h, k) = (1, -3)\) from the terms \((x - h)^2\) and \((y - k)^2\).

Calculate the focal distance \(c\) using the formula \(c^2 = a^2 - b^2\). Then, determine the coordinates of the foci by moving \(c\) units along the major axis (vertical) from the center.

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주요 개념

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

Standard Form of an Ellipse

An ellipse equation in standard form is written as (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 semi-major and semi-minor axes, respectively. Converting the given equation to this form is essential for graphing and analyzing the ellipse.

Identifying the Major and Minor Axes

The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. By comparing a² and b², you determine which axis is major (larger value) and which is minor (smaller value). This helps in correctly sketching the ellipse and locating its vertices.

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. Knowing the foci is crucial for understanding ellipse properties.