Graph each ellipse and locate the foci. x² = 1 – 4y²

검증된 단계별 안내

Rewrite the given equation \(x^2 = 1 - 4y^2\) to the standard form of an ellipse equation. Start by moving all terms to one side: \(x^2 + 4y^2 = 1\).

Divide both sides of the equation by 1 to express it in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, it becomes \(\frac{x^2}{1} + \frac{y^2}{\frac{1}{4}} = 1\).

Identify the values of \(a^2\) and \(b^2\). From the equation, \(a^2 = 1\) and \(b^2 = \frac{1}{4}\). Since \(a^2 > b^2\), the major axis is along the x-axis.

Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Substitute the values to get \(c^2 = 1 - \frac{1}{4}\).

Locate the foci at points \((\pm c, 0)\) on the x-axis. These points represent the foci of the ellipse. Finally, sketch the ellipse using the intercepts and foci.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

주요 개념

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

Standard Form of an Ellipse

An ellipse is typically expressed in the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h,k)\) is the center, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes. Understanding how to rewrite the given equation into this form is essential for graphing the ellipse.

Identifying the Orientation of the Ellipse

The ellipse can be oriented horizontally or vertically depending on whether the \(x^2\) or \(y^2\) term is associated with the larger denominator. Recognizing the orientation helps in correctly plotting the ellipse and locating its axes.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse, located along the major axis, found using \( c^2 = |a^2 - b^2| \). Knowing how to calculate \(c\) and place the foci relative to the center is crucial for completing the graph accurately.