Find the Foci (x^2)/4+(y^2)/5=1

Solution:

Step:1

Normalize the equation to have 1 on the right side, conforming to the standard form of an ellipse or hyperbola equation. The given equation is already in the desired form: $\frac{x^{2}}{4} + \frac{y^{2}}{5} = 1$.

Step:2

Recognize that the equation represents an ellipse. The general equation for an ellipse is $\frac{(x - h)^{2}}{b^{2}} + \frac{(y - k)^{2}}{a^{2}} = 1$, where $(h,k)$ is the center, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Step:3

Identify the values corresponding to $a$, $b$, $h$, and $k$ from the given equation, where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $(h,k)$ is the center of the ellipse. We find $a = \sqrt{5}$, $b = 2$, $h = 0$, and $k = 0$.

Step:4

Determine the distance $c$ from the center to a focus of the ellipse.

Step:4.1

Use the formula $c = \sqrt{a^{2} - b^{2}}$ to calculate the distance.

Step:4.2

Plug in the values for $a$ and $b$: $c = \sqrt{(\sqrt{5})^{2} - 2^{2}}$.

Step:4.3

Simplify the expression.

Step:4.3.1

Convert $(\sqrt{5})^{2}$ to $5$.

Step:4.3.1.1

Express $\sqrt{5}$ as $5^{\frac{1}{2}}$ and apply the formula: $c = \sqrt{(5^{\frac{1}{2}})^{2} - 2^{2}}$.

Step:4.3.1.2

Use the power rule to simplify: $c = \sqrt{5^{\frac{1}{2} \cdot 2} - 2^{2}}$.

Step:4.3.1.3

Combine the exponents: $c = \sqrt{5^{1} - 2^{2}}$.

Step:4.3.1.4

Simplify the expression: $c = \sqrt{5 - 2^{2}}$.

Step:4.3.2

Finalize the simplification.

Step:4.3.2.1

Square the number 2: $c = \sqrt{5 - 4}$.

Step:4.3.2.2

Subtract 4 from 5: $c = \sqrt{1}$.

Step:4.3.2.3

The square root of 1 is 1: $c = 1$.

Step:5

Locate the foci of the ellipse.

Step:5.1

To find the first focus, add $c$ to $k$: $(h, k + c)$.

Step:5.2

Insert the known values for $h$, $k$, and $c$: $(0, 0 + 1)$.

Step:5.3

The result simplifies to $(0, 1)$.

Step:5.4

To find the second focus, subtract $c$ from $k$: $(h, k - c)$.

Step:5.5

Insert the known values for $h$, $k$, and $c$: $(0, 0 - 1)$.

Step:5.6

The result simplifies to $(0, -1)$.

Step:5.7

An ellipse has two foci: $(\text{Focus})_{1}: (0, 1)$ and $(\text{Focus})_{2}: (0, -1)$.

Step:6

The foci of the ellipse are at $(0, 1)$ and $(0, -1)$.

Knowledge Notes:

1. Ellipse Equation: The standard form of an ellipse is $\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1$, where $(h,k)$ is the center of the ellipse, $a$ is the semi-major axis, and $b$ is the semi-minor axis.

2. Foci of an Ellipse: The foci of an ellipse are two fixed points located along the major axis, equidistant from the center. The distance from the center to each focus is given by $c = \sqrt{a^{2} - b^{2}}$.

3. Major and Minor Axes: In an ellipse, the major axis is the longest diameter, and the minor axis is the shortest diameter. The lengths of the semi-major axis and the semi-minor axis are denoted by $a$ and $b$, respectively.

4. Simplifying Square Roots: When simplifying expressions involving square roots, remember that $\sqrt{a^{2}} = a$ and $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$.

5. Finding Foci Coordinates: The coordinates of the foci of an ellipse can be found by adding and subtracting the value of $c$ from the $y$-coordinate of the center if the major axis is vertical, or from the $x$-coordinate if the major axis is horizontal. In this case, since $h$ and $k$ are both zero, the foci are at $(0, k + c)$ and $(0, k - c)$.