Find the Foci 9y^2-25x^2=225

Solution:

Step:1

Convert the equation to the standard form of a hyperbola.

Step:1.1

Normalize the equation by dividing all terms by $225$ to get the constant term on the right side to be one. $\frac{9y^2}{225} - \frac{25x^2}{225} = \frac{225}{225}$

Step:1.2

Reduce the equation to its simplest form to set the right side to $1$. The standard equation of a hyperbola requires the constant term to be $1$. $\frac{y^2}{25} - \frac{x^2}{9} = 1$

Step:2

Recognize the structure as a hyperbola. Use the standard equation to identify parameters for locating vertices and asymptotes. $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

Step:3

Align the given hyperbola's terms with the standard form. Here, $h$ is the horizontal shift, $k$ is the vertical shift, and $a$ and $b$ are constants. $a = 5$, $b = 3$, $k = 0$, $h = 0$

Step:4

Calculate $c$, the distance from the center to a focus of the hyperbola.

Step:4.1

Use the hyperbola's focal distance formula. $c = \sqrt{a^2 + b^2}$

Step:4.2

Insert the known values for $a$ and $b$ into the formula. $c = \sqrt{5^2 + 3^2}$

Step:4.3

Perform the calculations.

Step:4.3.1

Square $5$. $c = \sqrt{25 + 3^2}$

Step:4.3.2

Square $3$. $c = \sqrt{25 + 9}$

Step:4.3.3

Add the results. $c = \sqrt{34}$

Step:5

Determine the coordinates of the foci.

Step:5.1

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

Step:5.2

Input the values for $h$, $c$, and $k$. The first focus is $(0, \sqrt{34})$.

Step:5.3

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

Step:5.4

Input the values for $h$, $c$, and $k$. The second focus is $(0, -\sqrt{34})$.

Step:5.5

The general form for the foci of a hyperbola is $(h, k \pm c)$. There are two foci for a hyperbola. The foci are $(0, \sqrt{34})$ and $(0, -\sqrt{34})$.

Step:6

The foci of the hyperbola given by the equation $9y^2 - 25x^2 = 225$ are located at $(0, \sqrt{34})$ and $(0, -\sqrt{34})$.

Knowledge Notes:

• The standard form of a hyperbola with a vertical transverse axis is $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$, where $(h, k)$ is the center of the hyperbola, $a$ is the distance from the center to the vertices along the y-axis, and $b$ is the distance from the center to the vertices along the x-axis.

• The foci of a hyperbola are located at a distance $c$ from the center, where $c = \sqrt{a^2 + b^2}$.

• The coordinates of the foci for a hyperbola centered at the origin with a vertical transverse axis are $(0, \pm c)$.

• When dividing an equation by a number to normalize it, ensure that all terms including the constant are divided by the same number to maintain the equation's balance.

• Squaring a number means multiplying the number by itself, e.g., $5^2 = 5 \times 5 = 25$.

• The square root of a number $x$, denoted as $\sqrt{x}$, is a value that, when multiplied by itself, gives $x$.