To solve the equation $4x^2 - 9y^2 - 16x + 54y - 101 = 0$, we can try rewriting it by completing the square to identify its form and analyze it. Here is the step-by-step solution:

Step 1: Group the terms by $x$ and $y$

Separate the $x$-terms and $y$-terms: $4x^2 - 16x - 9y^2 + 54y = 101$

Step 2: Factor out coefficients of squared terms

Rewrite by factoring out the coefficients for $x$-terms and $y$-terms: $4(x^2 - 4x) - 9(y^2 - 6y) = 101$

Step 3: Complete the square for both $x$ and $y$

1. For $x$-terms: $x^2 - 4x$

   • Take half of $-4$, square it: $\left(-2\right)^2 = 4$
   • Add and subtract 4 inside the parentheses: $4(x^2 - 4x + 4 - 4) = 4((x - 2)^2 - 4)$
   • Which simplifies to $4(x - 2)^2 - 16$

2. For $y$-terms: $y^2 - 6y$

   • Take half of $-6$, square it: $\left(-3\right)^2 = 9$
   • Add and subtract 9 inside the parentheses: $-9(y^2 - 6y + 9 - 9) = -9((y - 3)^2 - 9)$
   • Which simplifies to $-9(y - 3)^2 + 81$

Substituting back, we get: $4(x - 2)^2 - 16 - 9(y - 3)^2 + 81 = 101$

Step 4: Simplify the equation

Combine constants: $4(x - 2)^2 - 9(y - 3)^2 + 65 = 101$ $4(x - 2)^2 - 9(y - 3)^2 = 36$

Step 5: Divide by 36

$\frac{4(x - 2)^2}{36} - \frac{9(y - 3)^2}{36} = 1$ $\frac{(x - 2)^2}{9} - \frac{(y - 3)^2}{4} = 1$

Conclusion

The equation is now in the form of a hyperbola: $\frac{(x - 2)^2}{3^2} - \frac{(y - 3)^2}{2^2} = 1$

This represents a hyperbola centered at $(2, 3)$, with a transverse axis along the $x$-axis. The semi-major axis length is 3 (in the $x$-direction), and the semi-minor axis length is 2 (in the $y$-direction).

Would you like a further breakdown of hyperbolas, or do you have specific questions?

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Here are five questions to extend your understanding:

1. How do we distinguish a hyperbola from an ellipse in standard form?
2. Why do we divide by 36 in the final step of simplifying?
3. What would change if the signs in front of $x^2$ and $y^2$ were the same?
4. Can you find the asymptotes for this hyperbola?
5. How would this equation look in polar coordinates?

Tip: Completing the square is a valuable technique for rewriting quadratic equations, particularly in identifying conic sections like circles, ellipses, and hyperbolas.