Hyperbola Equation Centered At Origin With Intercepts And Asymptote

Hey guys! Today, we're diving deep into the fascinating world of hyperbolas, specifically those centered at the origin. We'll tackle a classic problem: finding the equation of a hyperbola given its x-intercepts and asymptote. Buckle up, because we're about to unravel the mysteries of these intriguing curves!

Understanding Hyperbolas

Before we jump into the problem, let's refresh our understanding of hyperbolas. A hyperbola is a conic section formed by the intersection of a plane with a double cone. Think of it as two mirrored parabolas opening away from each other. The key features of a hyperbola centered at the origin are:

• Center: The point where the two axes of symmetry intersect (in our case, (0, 0)).
• Vertices: The points where the hyperbola intersects its transverse axis. These are the closest points on the hyperbola to the center.
• Foci: Two fixed points inside the hyperbola that define its shape. The difference in distances from any point on the hyperbola to the two foci is constant.
• Transverse Axis: The axis that passes through the vertices and foci. The distance between the vertices is 2a, where 'a' is a crucial parameter in the hyperbola's equation.
• Conjugate Axis: The axis perpendicular to the transverse axis, passing through the center. Its length is 2b, where 'b' is another vital parameter.
• Asymptotes: These are straight lines that the hyperbola approaches as it extends infinitely. They act as guidelines for the hyperbola's shape and are essential for sketching it accurately.

The standard equation of a hyperbola centered at the origin depends on whether it opens horizontally or vertically:

• Horizontal Hyperbola:

  $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

  In this case, the vertices are at (±a, 0), and the transverse axis lies along the x-axis.
• Vertical Hyperbola:

  $$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$

  Here, the vertices are at (0, ±a), and the transverse axis lies along the y-axis.

The relationship between a, b, and the slopes of the asymptotes is crucial. For a horizontal hyperbola, the asymptotes are given by the equations y = ±(b/a)x. For a vertical hyperbola, the asymptotes are y = ±(a/b)x. This connection between the hyperbola's parameters and its asymptotes is what we'll leverage to solve our problem.

Understanding these fundamental concepts is critical for tackling any hyperbola problem. We need to visualize the shape, identify the key parameters, and relate them to the given information. Now, let's get back to our specific challenge!

Problem Breakdown: Finding the Equation

Okay, let's break down the problem step-by-step. We're given a hyperbola centered at the origin with x-intercepts ±3 and an asymptote y = 2x. Our mission is to find the equation of this hyperbola.

First, the fact that the hyperbola has x-intercepts at ±3 immediately tells us that it's a horizontal hyperbola. Why? Because the x-intercepts are the points where the hyperbola crosses the x-axis, which lies along the transverse axis for a horizontal hyperbola. This is a crucial piece of information that narrows down our possibilities.

Secondly, the x-intercepts give us the value of 'a'. Remember, for a horizontal hyperbola, the vertices are at (±a, 0). Since the x-intercepts are ±3, we know that a = 3. This means a² = 9, which will be the denominator of the x² term in our equation.

Now, let's tackle the asymptote. We're given the asymptote y = 2x. For a horizontal hyperbola, the asymptotes are of the form y = ±(b/a)x. We already know a = 3, so we can set up the equation:

$$2 = \frac{b}{a} = \frac{b}{3}$$

Solving for b, we get:

$$b = 2 * 3 = 6$$

Therefore, b = 6, and b² = 36. This will be the denominator of the y² term in our equation.

With 'a' and 'b' determined, we have all the ingredients to write the equation of the hyperbola. We know it's a horizontal hyperbola, so the equation will be in the form:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

Substituting our values for a² and b², we get:

$$\frac{x^2}{9} - \frac{y^2}{36} = 1$$

And there you have it! We've successfully found the equation of the hyperbola. This systematic approach, breaking down the problem into smaller, manageable steps, is key to conquering these types of questions.

Putting It All Together: The Solution

So, let's recap. We were given a hyperbola centered at the origin with x-intercepts ±3 and an asymptote y = 2x. We deduced that it's a horizontal hyperbola because of the x-intercepts. We found 'a' to be 3 from the x-intercepts and 'b' to be 6 using the asymptote equation. This led us to the equation:

$$\frac{x^2}{9} - \frac{y^2}{36} = 1$$

Therefore, the correct answer is c. $\frac{x^{2}{9}-\frac{y}2}{36}=1$

This problem highlights the importance of understanding the properties of hyperbolas and how they relate to their equations. By recognizing the key features and applying the correct formulas, we can confidently solve these types of problems. Remember, practice makes perfect, so keep those hyperbola problems coming!

Diving Deeper: Asymptotes and Hyperbola Behavior

Now that we've nailed the solution, let's delve a little deeper into the significance of asymptotes in understanding hyperbola behavior. Asymptotes, guys, are like the guiding rails for a hyperbola. They dictate how the hyperbola curves and extends infinitely. The closer the hyperbola gets to infinity, the closer it hugs its asymptotes, but it never actually touches them. This unique behavior is a defining characteristic of hyperbolas.

The slopes of the asymptotes are directly linked to the ratio of 'a' and 'b', the parameters that define the hyperbola's shape. For a horizontal hyperbola, the slopes are ±(b/a), and for a vertical hyperbola, they are ±(a/b). This relationship allows us to visualize the hyperbola's shape and orientation just by knowing the asymptote equations.

Consider our problem again. The asymptote y = 2x tells us that the ratio b/a is 2. This means that 'b' is twice as large as 'a'. Geometrically, this implies that the hyperbola is