Graph the ellipse and locate the foci. x236+y225=1\frac{$$x^2$$}{36} + \frac{$$y^2$$}{25} = 1

Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

문제 1

8장, 문제 1

Graph the ellipse and locate the foci. $\frac{x^{2}}{36} + \frac{y^{2}}{25} = 1$

검증된 단계별 안내

Identify the standard form of the ellipse equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Here, $a^2 = 36$ and $b^2 = 25$.

Determine the values of $a$ and $b$ by taking the square roots: $a = \sqrt{36} = 6$ and $b = \sqrt{25} = 5$.

Since $a > b$, the major axis is along the x-axis. The ellipse is centered at the origin $(0,0)$ with vertices at $(\pm a, 0)$, which are $(\pm 6, 0)$.

Calculate the focal distance $c$ using the relationship $c^2 = a^2 - b^2$. Substitute the values to find $c^2 = 36 - 25$.

Locate the foci at $(\pm c, 0)$ along the x-axis. These points are inside the ellipse between the center and the vertices.

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주요 개념

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

Standard Form of an Ellipse

An ellipse in standard form is written as (x^2/a^2) + (y^2/b^2) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Identifying a and b helps determine the shape and orientation of the ellipse on the coordinate plane.

Graphing an Ellipse

To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connecting these points smoothly forms the ellipse, showing its size and orientation.

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 c, found using c^2 = a^2 - b^2. Knowing c allows you to place the foci accurately on the graph.

Graph the ellipse and locate the foci. x236+y225=1\(\frac{x^2}{36}\) + \(\frac{y^2}{25}\) = 1