Find the Foci (x^2)/36-(y^2)/49=1

Solution:

Step 1:

Rewrite the given equation to conform to the standard form where the right side equals $1$. The equation of an ellipse or hyperbola in standard form should have a right side equal to $1$.

$$\frac{x^{2}}{36} - \frac{y^{2}}{49} = 1$$

Step 2:

Recognize the equation as that of a hyperbola. The standard form of a hyperbola is used to identify the parameters needed to locate vertices and asymptotes.

$$\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$$

Step 3:

Align the given hyperbola's coefficients with the standard form's variables. Here, $h$ is the horizontal shift from the origin, $k$ is the vertical shift, and $a$ and $b$ are the denominators under $x^{2}$ and $y^{2}$ respectively.

$$a = 6$$ $$b = 7$$ $$k = 0$$ $$h = 0$$

Step 4:

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

Step 4.1:

Use the hyperbola's focal distance formula to calculate $c$.

$$c = \sqrt{a^{2} + b^{2}}$$

Step 4.2:

Insert the known values for $a$ and $b$ into the formula.

$$c = \sqrt{6^{2} + 7^{2}}$$

Step 4.3:

Carry out the simplification process.

Step 4.3.1:

Square $6$.

$$c = \sqrt{36 + 7^{2}}$$

Step 4.3.2:

Square $7$.

$$c = \sqrt{36 + 49}$$

Step 4.3.3:

Combine $36$ and $49$.

$$c = \sqrt{85}$$

Step 5:

Locate the foci of the hyperbola.

Step 5.1:

To find the first focus, add $c$ to $h$.

$$(h + c, k)$$

Step 5.2:

Substitute the values for $h$, $c$, and $k$ and simplify.

$$(\sqrt{85}, 0)$$

Step 5.3:

To find the second focus, subtract $c$ from $h$.

$$(h - c, k)$$

Step 5.4:

Substitute the values for $h$, $c$, and $k$ and simplify.

$$(-\sqrt{85}, 0)$$

Step 5.5:

The foci of a hyperbola are given by the formula $(h \pm c, k)$. There are two foci for a hyperbola.

$$(\sqrt{85}, 0), (-\sqrt{85}, 0)$$

Step 6:

The foci of the hyperbola are at the points $(\sqrt{85}, 0)$ and $(-\sqrt{85}, 0)$.

Knowledge Notes:

1. Standard Form of a Hyperbola: The equation of a hyperbola in standard form is $\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1$ for a horizontal hyperbola, and $\frac{(y - k)^{2}}{b^{2}} - \frac{(x - h)^{2}}{a^{2}} = 1$ for a vertical hyperbola, where $(h, k)$ is the center of the hyperbola, $a$ is the distance from the center to the vertices along the transverse axis, and $b$ is the distance from the center to the vertices along the conjugate axis.

2. Foci of a Hyperbola: The foci of a hyperbola are located at a distance $c$ from the center, where $c = \sqrt{a^{2} + b^{2}}$. For a horizontal hyperbola, the foci are at $(h \pm c, k)$, and for a vertical hyperbola, they are at $(h, k \pm c)$.

3. Vertices of a Hyperbola: The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis. For a horizontal hyperbola, the vertices are at $(h \pm a, k)$, and for a vertical hyperbola, they are at $(h, k \pm a)$.

4. Asymptotes of a Hyperbola: The asymptotes of a hyperbola are straight lines that the hyperbola approaches but never touches. The equations of the asymptotes for a hyperbola centered at $(h, k)$ are $y = k \pm \frac{b}{a}(x - h)$ for a horizontal hyperbola and $x = h \pm \frac{a}{b}(y - k)$ for a vertical hyperbola.

5. Simplifying Square Roots: When simplifying square roots, combine the terms under the radical if possible, and then take the square root of any perfect squares to simplify the expression further.