Find the Foci y^2+2y=9x^2+8

Solution:

Step 1: Convert the equation to the standard form of a hyperbola.

Step 1.1: Isolate the variable terms on one side.

• Subtract $9x^2$ from both sides: $y^2 + 2y - 9x^2 = 8$
• Rearrange the terms: $y^2 - 9x^2 + 2y = 8$

Step 1.2: Complete the square for the $y$ terms.

• Identify the coefficients: $a = 1$, $b = 2$, $c = 0$
• Use the vertex form $a(x + d)^2 + e$
• Calculate $d = \frac{b}{2a}$: $d = \frac{2}{2 \cdot 1} = 1$
• Calculate $e = c - \frac{b^2}{4a}$: $e = 0 - \frac{2^2}{4 \cdot 1} = -1$
• Substitute $d$ and $e$ into the vertex form: $(y + 1)^2 - 1$

Step 1.3: Replace $y^2 + 2y$ with the completed square in the equation.

• $-9x^2 + (y + 1)^2 - 1 = 8$

Step 1.4: Add $1$ to both sides to balance the equation.

• $-9x^2 + (y + 1)^2 = 9$

Step 1.5: Simplify the equation by adding constants on the right side.

• $-9x^2 + (y + 1)^2 = 9$

Step 1.6: Divide all terms by $9$ to normalize the right side to $1$.

• $-\frac{x^2}{1} + \frac{(y + 1)^2}{9} = 1$

Step 1.7: The equation is now in standard form.

• $\frac{(y + 1)^2}{9} - \frac{x^2}{1} = 1$

Step 2: Use the standard form to identify the hyperbola's parameters.

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

Step 3: Match the equation's coefficients with the standard form.

• $a = 3$, $b = 1$, $k = -1$, $h = 0$

Step 4: Determine the distance to the foci ($c$).

Step 4.1: Use the formula $c = \sqrt{a^2 + b^2}$.

Step 4.2: Substitute $a$ and $b$ into the formula.

• $c = \sqrt{3^2 + 1^2}$

Step 4.3: Simplify the expression.

• $c = \sqrt{9 + 1} = \sqrt{10}$

Step 5: Locate the foci of the hyperbola.

Step 5.1: The first focus is found by adding $c$ to $k$.

• Focus 1: $(h, k + c)$

Step 5.2: Substitute the values and simplify.

• Focus 1: $(0, -1 + \sqrt{10})$

Step 5.3: The second focus is found by subtracting $c$ from $k$.

• Focus 2: $(h, k - c)$

Step 5.4: Substitute the values and simplify.

• Focus 2: $(0, -1 - \sqrt{10})$

Step 5.5: The hyperbola's foci are given by $(h, k \pm c)$.

• Foci: $(0, -1 + \sqrt{10}), (0, -1 - \sqrt{10})$

Knowledge Notes:

The process of finding the foci of a hyperbola involves several key steps and knowledge points:

1. Standard Form of a Hyperbola: 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, and $b$ is the distance from the center to the co-vertices.

2. Completing the Square: This technique is used to transform a quadratic expression into a perfect square trinomial, which helps in rewriting the equation in standard form.

3. Vertex Form of a Parabola: The vertex form is $a(x - h)^2 + k$, where $(h, k)$ is the vertex. For a hyperbola, we use a similar approach for the $y$ terms.

4. Foci of a Hyperbola: The foci are located along the transverse axis, a distance $c$ from the center, where $c = \sqrt{a^2 + b^2}$.

5. Hyperbola Orientation: The sign before the $x^2$ term indicates whether the hyperbola opens up and down (positive sign) or left and right (negative sign). In this case, the negative sign indicates a vertical transverse axis.

6. Latex Formatting: Mathematical expressions are formatted using Latex to clearly present the steps and calculations involved in solving the problem.