Identifying Hyperbola Equations With Furthest Foci

by Felix Dubois 51 views

In the fascinating world of mathematics, particularly in analytic geometry, hyperbolas hold a special place. These conic sections, defined by their unique properties and equations, pop up in various real-world applications, from the trajectories of comets to the design of cooling towers. Today, guys, we're diving deep into the realm of hyperbolas, focusing on how to identify the equation representing a hyperbola with foci located the farthest from its center. Buckle up, because this journey involves understanding the key parameters of a hyperbola's equation and how they relate to its geometry.

Understanding Hyperbolas

Before we tackle the question directly, let's refresh our understanding of hyperbolas. A hyperbola is defined as the set of all points where the difference of the distances to two fixed points (the foci) is constant. This definition gives rise to the characteristic two-branch shape of the hyperbola. The standard form equation of a hyperbola centered at (h, k) depends on whether the hyperbola opens horizontally or vertically:

  • Horizontal Hyperbola: ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
  • Vertical Hyperbola: ((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1

Key parameters in these equations are 'a' and 'b'. 'a' represents the distance from the center to each vertex along the transverse axis (the axis that passes through the foci and vertices), and 'b' is related to the distance along the conjugate axis. The distance from the center to each focus, denoted by 'c', is given by the relationship: c^2 = a^2 + b^2. This relationship is crucial because it directly connects the parameters of the equation to the distance of the foci from the center.

Foci Distance and the Equation

The question at hand asks us to determine which hyperbola equation represents the hyperbola with foci farthest from its center. This means we need to find the hyperbola with the largest 'c' value. Looking at the equation c^2 = a^2 + b^2, we can see that 'c' increases as either 'a' or 'b' (or both) increase. In other words, the larger the values of a^2 and b^2 in the hyperbola's equation, the farther the foci will be from the center. So, our strategy will be to calculate 'c' for each given option and compare the values. This involves carefully extracting 'a' and 'b' from each equation, applying the formula, and then determining which 'c' is the largest. It’s like a mathematical treasure hunt, where we’re searching for the biggest ‘c’ treasure!

Analyzing the Given Options

Let's dive into analyzing the given options, guys. We need to extract the values of 'a' and 'b' from each equation, calculate 'c', and then compare the 'c' values to determine which hyperbola has foci farthest from the center. It's like being a detective, but instead of clues, we have equations! Let's put on our mathematical detective hats and get to work.

Option A: ((x - 2)^2 / 8^2) - ((y - 1)^2 / 7^2) = 1

In this equation, we can readily identify that a^2 = 8^2 = 64 and b^2 = 7^2 = 49. Notice how the equation is already in the standard form, making it easy for us to pluck out these values. Now, let's calculate c^2 using the formula c^2 = a^2 + b^2:

c^2 = 64 + 49 = 113

Therefore, c = √113. This is our first data point. We'll keep this value in mind as we analyze the other options. Think of it as setting a benchmark in our quest for the largest ‘c’.

Option B: ((2y + 4)^2 / 19^2) - ((2x - 6)^2 / 11^2) = 1

This equation looks a bit trickier, doesn't it? We need to be careful here. Notice the coefficients '2' within the parentheses. To get this into standard form, we need to factor out the '2' from both terms. Remember, the standard form requires the x and y terms to have a coefficient of 1. This is a common pitfall, so it's important to pay close attention to these details. Factoring out the '2', we get:

((2(y + 2))^2 / 19^2) - ((2(x - 3))^2 / 11^2) = 1

This simplifies to:

(4(y + 2)^2 / 19^2) - (4(x - 3)^2 / 11^2) = 1

Now, we need to divide both the numerator and denominator by 4 to further simplify and isolate the squared terms:

((y + 2)^2 / (19^2 / 4)) - ((x - 3)^2 / (11^2 / 4)) = 1

Thus, a^2 = 19^2 / 4 = 361 / 4 = 90.25 and b^2 = 11^2 / 4 = 121 / 4 = 30.25. See how the seemingly small coefficients made a significant difference in our calculations? This is why meticulousness is key in mathematics!

Now we calculate c^2:

c^2 = 90.25 + 30.25 = 120.5

So, c = √120.5. This is a higher value than what we found in Option A! Our treasure hunt is getting exciting.

Comparing the Results and Conclusion

Alright, guys, we've done the heavy lifting! We've calculated the 'c' values for the given options. Now, it's time to compare and declare our winner – the hyperbola with the foci farthest from the center. Let's put our results side-by-side:

  • Option A: c = √113 ≈ 10.63
  • Option B: c = √120.5 ≈ 10.98

By comparing the values, we can clearly see that the hyperbola represented by Option B has a larger 'c' value (approximately 10.98) than the hyperbola in Option A (approximately 10.63). Therefore, the foci of the hyperbola in Option B are farther from its center.

In conclusion, Option B represents the hyperbola with foci farthest from the center. This exercise highlights the importance of understanding the standard form equation of a hyperbola and how the parameters 'a' and 'b' influence the distance 'c' from the center to the foci. Remember, when dealing with hyperbolas, pay close attention to the coefficients and always aim to transform the equation into its standard form for easy analysis. And most importantly, keep practicing and exploring the fascinating world of conic sections! You've got this!