Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

검증된 단계별 안내

Identify the center of the ellipse. Since the foci are at (-2, 0) and (2, 0), the center is the midpoint of the foci, which is at (0, 0).

Determine the orientation of the ellipse. The foci lie on the x-axis, so the major axis is horizontal.

Calculate the distance between the center and each focus, denoted as \(c\). Here, \(c = 2\) because the foci are at \(\pm 2\) on the x-axis.

Use the y-intercepts to find the length of the minor axis. The y-intercepts are at \(y = \pm 3\), so the minor axis length is \(2b = 6\), giving \(b = 3\).

Apply the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^2 = a^2 - b^2\). Use this to solve for \(a^2\), then write the standard form equation of the ellipse centered at the origin with a horizontal major axis: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

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 depends on the orientation of its major axis. For a horizontal major axis centered at the origin, it is (x²/a²) + (y²/b²) = 1, where 'a' is the semi-major axis and 'b' is the semi-minor axis. Understanding this form helps in writing the ellipse equation once 'a' and 'b' are known.

Foci and Relationship to Axes

The foci of an ellipse are two fixed points used to define the curve. The distance from the center to each focus is 'c', and it relates to the axes by the equation c² = a² - b² for horizontal ellipses. Knowing the foci helps determine 'c', which is essential to find 'a' and 'b'.

Intercepts and Their Role in Determining Axis Lengths

The y-intercepts of an ellipse give the points where the ellipse crosses the y-axis, indicating the length of the minor axis. Since the ellipse crosses the y-axis at ±b, the intercepts help find the value of 'b'. This information, combined with 'c', allows solving for 'a' and writing the equation.