Graph each ellipse and locate the foci. 25x²+4y² = 100

검증된 단계별 안내

Rewrite the given equation of the ellipse in standard form by dividing both sides of the equation by 100: \(\frac{25x^{2}}{100} + \frac{4y^{2}}{100} = \frac{100}{100}\).

Simplify the fractions to get the equation in the form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).

Identify the values of \(a^{2}\) and \(b^{2}\) from the simplified equation. Determine which is larger to know the orientation of the ellipse (horizontal or vertical).

Calculate the value of \(c\) using the relationship \(c^{2} = |a^{2} - b^{2}|\), where \(c\) is the distance from the center to each focus.

Locate the foci on the coordinate plane by placing them \(c\) units along the major axis from the center at the origin, and sketch the ellipse using the intercepts from \(a\) and \(b\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

주요 개념

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

Standard Form of an Ellipse

An ellipse equation can be written in the standard form (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form helps identify these axes and facilitates graphing the ellipse.

Identifying the Major and Minor Axes

The major axis is the longer axis of the ellipse, and the minor axis is the shorter one. By comparing a² and b² from the standard form, you determine which axis is major or minor, which is essential for correctly sketching the ellipse and locating its foci.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse located along the major axis. Their distance from the center is given by c, where c² = |a² - b²|. Knowing a, b, and c allows you to plot the foci accurately on the graph.