Curve Identification: Circle, Ellipse, Hyperbola, Or Parabola?

by TheNnagam 63 views

Alright guys, let's dive into the exciting world of conic sections! Today, we're tackling a classic problem in analytic geometry: identifying the type of curve represented by a given equation. Specifically, we need to figure out whether the equation X² + Y² + 6X - 2Y - 65 = 0 represents a circle, an ellipse, a hyperbola, or a parabola. Buckle up, because we're about to embark on a mathematical adventure!

Understanding Conic Sections

Before we jump into the specifics of our equation, it's super helpful to have a solid understanding of what each of these curves actually is. Think of it as knowing the players before the game starts.

  • Circle: A circle is the set of all points equidistant from a central point. In simpler terms, it's a round shape where every point on the edge is the same distance from the middle. The general equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  • Ellipse: An ellipse is like a stretched-out circle. Instead of having a single radius, it has two: a major axis (the longer one) and a minor axis (the shorter one). The general equation of an ellipse centered at the origin is x²/a² + y²/b² = 1, where 'a' is the semi-major axis and 'b' is the semi-minor axis. If a = b, you guessed it, you have a circle!
  • Hyperbola: A hyperbola is a curve consisting of two symmetrical branches that open away from each other. It's defined by the difference of distances from two foci being constant. The general equation of a hyperbola centered at the origin is x²/a² - y²/b² = 1 (horizontal opening) or y²/a² - x²/b² = 1 (vertical opening).
  • Parabola: A parabola is a U-shaped curve, defined as the set of all points equidistant from a focus point and a directrix line. The general equation of a parabola is y = ax² + bx + c (vertical opening) or x = ay² + by + c (horizontal opening).

Analyzing the Equation: X² + Y² + 6X - 2Y - 65 = 0

Now that we've refreshed our memory on conic sections, let's get back to the equation at hand: X² + Y² + 6X - 2Y - 65 = 0. The key to identifying the curve lies in manipulating this equation into a recognizable standard form.

The first thing we'll do is complete the square for both the x and y terms. Completing the square is a technique that allows us to rewrite a quadratic expression in the form of a squared term plus a constant. This will help us reveal the underlying structure of the equation.

Completing the Square

  1. Group x and y terms: Rearrange the equation to group the x terms together and the y terms together: (X² + 6X) + (Y² - 2Y) = 65.
  2. Complete the square for x: To complete the square for X² + 6X, we take half of the coefficient of the x term (which is 6), square it (which is 9), and add it to both sides of the equation. This gives us (X² + 6X + 9) + (Y² - 2Y) = 65 + 9.
  3. Complete the square for y: Similarly, to complete the square for Y² - 2Y, we take half of the coefficient of the y term (which is -2), square it (which is 1), and add it to both sides of the equation. This gives us (X² + 6X + 9) + (Y² - 2Y + 1) = 65 + 9 + 1.
  4. Rewrite as squared terms: Now, we can rewrite the expressions in parentheses as squared terms: (X + 3)² + (Y - 1)² = 75.

Identifying the Curve

Alright, check this out. We've transformed the original equation into the form (X + 3)² + (Y - 1)² = 75. Do you notice any similarities to the general equation of a circle? That's right, guys! This equation perfectly matches the standard form of a circle: (x - h)² + (y - k)² = r².

In our case, we have:

  • Center: (-3, 1)
  • Radius squared: r² = 75
  • Radius: r = √75 = 5√3

Therefore, the equation X² + Y² + 6X - 2Y - 65 = 0 represents a circle with a center at (-3, 1) and a radius of 5√3. Cool, right?

Why It's Not the Other Conic Sections

To further solidify our understanding, let's quickly discuss why the equation doesn't represent an ellipse, a hyperbola, or a parabola.

  • Ellipse: An ellipse has the general form x²/a² + y²/b² = 1 (centered at the origin). While our equation involves squared terms of both x and y, the coefficients of these terms are equal after completing the square (both are 1). In an ellipse, these coefficients would be different.
  • Hyperbola: A hyperbola has the general form x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1. The key characteristic of a hyperbola is the subtraction sign between the x² and y² terms. Our equation has an addition sign, so it cannot be a hyperbola.
  • Parabola: A parabola involves only one squared term (either x² or y²), while the other variable is linear. Our equation has both x² and y² terms, so it cannot be a parabola.

Conclusion

So, there you have it! By completing the square and comparing the resulting equation to the standard forms of conic sections, we've confidently identified that the equation X² + Y² + 6X - 2Y - 65 = 0 represents a circle. Remember, guys, practice makes perfect! The more you work with these equations, the easier it will become to recognize the different types of curves. Keep exploring, keep learning, and most importantly, keep having fun with math!

Key takeaways:

  • Completing the square is a powerful technique for identifying conic sections.
  • The standard form of a conic section equation reveals its key properties (center, radius, axes, etc.).
  • Understanding the general forms of ellipses, hyperbolas and parabolas helps you distinguish these from circles.

Now you're ready to tackle more challenging curve identification problems. Happy solving!

Practice Problems

Identify the curve represented by the following equations:

  1. 4x² + 9y² - 16x + 18y - 11 = 0
  2. y² - 8x - 6y + 25 = 0
  3. x² - y² + 2x - 2y - 1 = 0