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.