Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

검증된 단계별 안내

Identify the center of the ellipse by finding the midpoint of the vertices. Since the vertices are at (0, -6) and (0, 6), the center is at (0, 0).

Determine the orientation of the ellipse. The vertices lie on the y-axis, so the major axis is vertical. This means the standard form of the ellipse equation is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \(a\) is the distance from the center to a vertex along the y-axis.

Calculate \(a\), the distance from the center to a vertex. Since the vertices are at (0, ±6), \(a = 6\). So, \(a^2 = 36\).

Use the point (2, 4) that lies on the ellipse to find \(b^2\). Substitute \(x = 2\), \(y = 4\), and \(a^2 = 36\) into the ellipse equation: \( \frac{2^2}{b^2} + \frac{4^2}{36} = 1 \).

Solve the equation from step 4 for \(b^2\) by isolating \( \frac{4}{b^2} \), then multiply both sides by \(b^2\) and solve for \(b^2\). This will give you the value needed to write the standard form of the ellipse.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

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 vertical major axis centered at the origin, the equation is (x^2 / b^2) + (y^2 / a^2) = 1, where 'a' is the distance from the center to a vertex along the y-axis, and 'b' is the distance along the x-axis. Understanding this form helps in identifying the ellipse's shape and size.

Vertices of an Ellipse

Vertices are the points on the ellipse farthest from the center along the major axis. Given vertices at (0, -6) and (0, 6), the center is at the origin (0,0), and the length of the major axis is 12, so a = 6. Knowing the vertices allows determination of 'a' in the ellipse equation.

Using a Point to Find the Ellipse Parameters

Substituting a known point on the ellipse into the standard form equation allows solving for the unknown parameter 'b'. Here, the point (2, 4) lies on the ellipse, so plugging in x=2 and y=4 helps find 'b', completing the equation. This step is essential to fully define the ellipse.