Graph each ellipse and locate the foci. 7x² = 35-5y²

검증된 단계별 안내

Rewrite the given equation to standard form of an ellipse. Start with the equation: \(7x^{2} = 35 - 5y^{2}\). Move all terms to one side to get \(7x^{2} + 5y^{2} = 35\).

Divide every term by 35 to normalize the equation: \(\frac{7x^{2}}{35} + \frac{5y^{2}}{35} = \frac{35}{35}\), which simplifies to \(\frac{x^{2}}{5} + \frac{y^{2}}{7} = 1\).

Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Here, \(a^{2} = 5\) and \(b^{2} = 7\). Since \(b^{2} > a^{2}\), the major axis is vertical.

Calculate the focal distance \(c\) using the formula \(c^{2} = b^{2} - a^{2}\). Substitute the values to get \(c^{2} = 7 - 5\).

Locate the foci on the graph along the major axis (the y-axis) at points \((0, \pm c)\). Then, sketch the ellipse centered at the origin with vertices at \((0, \pm \sqrt{7})\) and co-vertices at \((\pm \sqrt{5}, 0)\).

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영상 길이:

11m

주요 개념

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

Standard Form of an Ellipse

An ellipse can be expressed in the standard form as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form is essential for graphing and identifying key features.

Identifying the Center and Axes Lengths

After rewriting the ellipse equation in standard form, determine the center coordinates and the values of a and b. These values represent the distances from the center to the ellipse's vertices along the major and minor axes, which are crucial for accurate graphing.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse, located along the major axis, defined by the distance c from the center, where c² = |a² - b²|. Knowing how to calculate and plot the foci helps in understanding the ellipse's geometric properties.