find the foci, vertex of y − x =1/y + x


To find the foci and vertices of the given equation \( y - x = \frac{1}{y + x} \), we must first manipulate it into a form that we can recognize as an ellipse, hyperbola, or parabola equation.

### Step 1: Substitute variables
Let:
\[
u = y - x \quad \text{and} \quad v = y + x.
\]
Then the equation becomes:
\[
u = \frac{1}{v}.
\]

### Step 2: Rewrite the equation
Multiplying both sides by \(v\) to eliminate the fraction:
\[
uv = 1.
\]
This is a hyperbolic equation. It can be recognized as a hyperbola of the form \( xy = 1 \), but in terms of the new variables \(u\) and \(v\).

### Step 3: Transform back to \(x\) and \(y\)
Now, reverse the substitutions:
\[
u = y - x \quad \text{and} \quad v = y + x.
\]
So the equation becomes:
\[
(y - x)(y + x) = 1.
\]
Using the difference of squares formula:
\[
y^2 - x^2 = 1.
\]
This is the equation of a hyperbola.

### Step 4: Standard form of the hyperbola
The equation \( y^2 - x^2 = 1 \) is already in the standard form of a hyperbola:
\[
\frac{y^2}{1} - \frac{x^2}{1} = 1,
\]
which represents a hyperbola centered at \( (0, 0) \), opening along the \(y\)-axis.

### Step 5: Find the vertices and foci
- **Vertices**: The vertices of the hyperbola are at \( (0, \pm 1) \), because the equation involves \( y^2 \) with a coefficient of 1.
- **Foci**: The foci are found using the formula \( c^2 = a^2 + b^2 \), where \( a = 1 \) and \( b = 1 \) (since the denominators in the hyperbola's standard form are both 1).
  
  Therefore:
  \[
  c^2 = 1 + 1 = 2 \quad \Rightarrow \quad c = \sqrt{2}.
  \]
  The foci are located at \( (0, \pm \sqrt{2}) \).

### Final Answer:
- **Vertices**: \( (0, 1) \) and \( (0, -1) \).
- **Foci**: \( (0, \sqrt{2}) \) and \( (0, -\sqrt{2}) \).

Would you like further details or explanations? Here are some additional questions you could explore:

1. How is the equation for a hyperbola derived from its geometric definition?
2. Can you find the asymptotes of the hyperbola \( y^2 - x^2 = 1 \)?
3. What are the parametric equations for this hyperbola?
4. How do you differentiate between equations of ellipses and hyperbolas?
5. Can you solve for the eccentricity of this hyperbola?

**Tip**: The foci of a hyperbola are always further from the center than the vertices, as the distance between them is determined by \( c = \sqrt{a^2 + b^2} \).

find the foci, vertex of y − x = 1/(y + x)

Explore how to find the foci and vertices of the hyperbola y^2 - x^2 = 1, with step-by-step solutions. This problem involves key concepts in conic sections, including hyperbolas, standard forms, and distance between foci. Suitable for high school students in Grades 10-12.

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