144y^2-25x^2+864y-50x-2329=0 sketch the graph of this in simple way please

To sketch the graph of the given equation:

\[
144y^2 - 25x^2 + 864y - 50x - 2329 = 0,
\]

we first recognize that this equation resembles the general form of a hyperbola. We need to rewrite it in a more standard form to identify its features and sketch it easily.

### Step 1: Simplify the Equation

Let's rewrite the equation by completing the square for both \(x\) and \(y\) terms.

#### 1. Group the \(x\) and \(y\) terms:

\[
144y^2 + 864y - 25x^2 - 50x = 2329.
\]

#### 2. Complete the square for the \(y\) terms:

Factor out \(144\):

\[
144(y^2 + 6y) = 144\left((y + 3)^2 - 9\right).
\]

So, the expression becomes:

\[
144(y + 3)^2 - 1296.
\]

#### 3. Complete the square for the \(x\) terms:

Factor out \(-25\):

\[
-25(x^2 + 2x) = -25\left((x + 1)^2 - 1\right).
\]

So, the expression becomes:

\[
-25(x + 1)^2 + 25.
\]

#### 4. Rewrite the equation:

\[
144(y + 3)^2 - 25(x + 1)^2 - 1296 + 25 = 2329.
\]

Simplify the constants:

\[
144(y + 3)^2 - 25(x + 1)^2 = 3600.
\]

#### 5. Divide the entire equation by \(3600\):

\[
\frac{(y + 3)^2}{25} - \frac{(x + 1)^2}{144} = 1.
\]

### Step 2: Identify the Graph

The simplified equation is a standard form of a hyperbola:

\[
\frac{(y + 3)^2}{25} - \frac{(x + 1)^2}{144} = 1,
\]

which is a hyperbola centered at \((-1, -3)\) with:

- Vertical transverse axis (since the \((y + 3)^2\) term is positive).
- Semi-major axis length of 5 (vertical direction).
- Semi-minor axis length of 12 (horizontal direction).

### Step 3: Sketching the Graph

1. **Plot the center** at \((-1, -3)\).
2. **Draw the box** defined by the semi-major and semi-minor axes lengths.
3. **Sketch the asymptotes** through the diagonals of the box.
4. **Draw the hyperbola** opening vertically, passing through the vertices along the \(y\)-axis directions.

Would you like me to provide a sketch or is the explanation sufficient?

144y^2 - 25x^2 + 864y - 50x - 2329 = 0 sketch the graph of this in simple way please

Discover how to sketch the graph of the hyperbola defined by the equation 144y^2 - 25x^2 + 864y - 50x - 2329 = 0. This step-by-step guide covers key concepts such as completing the square and analyzing hyperbolas. Suitable for students in Grades 10-12.

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