Graph the ellipse and locate the foci. (x^2)/36 +(y^2)/25 = 1

Verified step by step guidance

1. Identify the standard form of the ellipse equation: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Here, $a^2=36$ and $b^2=25$.
2. Determine the values of $a$ and $b$ by taking the square roots: $a=6$ and $b=5$.
3. Since $a>b$, the major axis is along the x-axis. The vertices are at $(\pm a,0)=(\pm 6,0)$.
4. Calculate the distance to the foci using $c^2=a^2-b^2$. Substitute the values: $c^2=36-25$.
5. Find $c$ by taking the square root of $c^2$, and locate the foci at $(\pm c,0)$.

Key Concepts

Ellipse Definition

An ellipse is a set of points in a plane where the sum of the distances from two fixed points, called foci, is constant. The standard form of an ellipse's equation is (x^2/a^2) + (y^2/b^2) = 1, where 'a' and 'b' are the semi-major and semi-minor axes, respectively. In this case, the ellipse is centered at the origin with axes aligned to the coordinate axes.

Graphing an Ellipse

To graph an ellipse, identify the lengths of the semi-major and semi-minor axes from the equation. For the given equation, a^2 = 36 (so a = 6) and b^2 = 25 (so b = 5). The ellipse will be elongated along the x-axis, with vertices at (±6, 0) and (0, ±5) on the y-axis, forming a closed curve.

Foci of an Ellipse

The foci of an ellipse are located along the major axis, at a distance 'c' from the center, where c is calculated using the formula c = √(a^2 - b^2). For this ellipse, a = 6 and b = 5, so c = √(36 - 25) = √11. Thus, the foci are located at (±√11, 0), which are approximately (±3.32, 0).