Finding The Foci Of A Hyperbola Solving (z-14)^2/16^3 - (y-2)^2/30^3 = 1

In this article, we will delve into the intricacies of hyperbolas, specifically focusing on how to determine the foci of a hyperbola given its equation. Our primary focus will be on dissecting the equation $\frac{(z-14)^2}{163}-\frac{(y-2)^2}{303}=1$ and identifying the correct foci from the provided options. Understanding the properties of hyperbolas, including their standard form equations, vertices, and the relationship between their parameters, is crucial for solving such problems. Let's embark on this mathematical journey to master the art of finding hyperbola foci.

Understanding Hyperbolas: A Foundation

Before we dive into the specifics of the given equation, let's establish a solid foundation by understanding the fundamental properties of hyperbolas. A hyperbola is a conic section formed by the intersection of a double cone and a plane that intersects both halves of the cone. It is defined as the set of all points in a plane such that the difference of the distances from two fixed points, called the foci, is constant. This constant difference is equal to the length of the transverse axis, which is the axis that passes through the foci and the vertices of the hyperbola.

The standard form of the equation of a hyperbola with a horizontal transverse axis is:

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Where:

• (h, k) is the center of the hyperbola.
• a is the distance from the center to each vertex along the transverse axis.
• b is the distance from the center to each vertex along the conjugate axis (the axis perpendicular to the transverse axis).

The distance from the center to each focus is denoted by 'c', and it is related to 'a' and 'b' by the equation:

$$c^2 = a^2 + b^2$$

The foci are located at (h ± c, k).

For a hyperbola with a vertical transverse axis, the standard form equation is:

$$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$$

In this case, the foci are located at (h, k ± c), and the relationship between a, b, and c remains the same: $c^2 = a^2 + b^2$. Distinguishing between horizontal and vertical hyperbolas is essential for correctly identifying the foci.

Analyzing the Given Equation: $\frac{(z-14)^2}{163}-\frac{(y-2)^2}{303}=1$

Now, let's turn our attention to the specific equation provided: $\frac{(z-14)^2}{163}-\frac{(y-2)^2}{303}=1$. This equation represents a hyperbola, and our goal is to determine the coordinates of its foci. By carefully examining the equation, we can extract key information necessary for finding the solution. The equation is in a form similar to the standard equation of a hyperbola with a horizontal transverse axis, but with 'z' instead of 'x' as the horizontal coordinate. This means the hyperbola opens along the horizontal direction (z-axis in this case).

Comparing the given equation with the standard form $\frac{(x-h)^2}{a2} - \frac{(y-k)^2}{b2} = 1$, we can identify the following parameters:

• The center of the hyperbola (h, k) is (14, 2).

• a^2 = 16^3 = 4096$, so $a = \sqrt{4096} = 64$.

• b^2 = 30^3 = 27000$, so $b = \sqrt{27000} = 30\sqrt{30}$.

The next crucial step is to calculate the value of 'c', which represents the distance from the center to each focus. We use the relationship $c^2 = a^2 + b^2$:

$$c^2 = 4096 + 27000 = 31096$$

Therefore, $c = \sqrt{31096} = \sqrt{16 \cdot 1943.5} = 4\sqrt{1943.5} \approx 176.34$.

Since the hyperbola has a horizontal transverse axis, the foci will be located at (h ± c, k). Substituting the values we found:

• Focus 1: (14 - 176.34, 2) ≈ (-162.34, 2)
• Focus 2: (14 + 176.34, 2) ≈ (190.34, 2)

These calculated coordinates do not directly match any of the provided options (A, B, C, or D). This indicates there might be a simplification or approximation involved in the answer choices, or perhaps an error in the initial calculation that needs to be revisited. Let's re-examine our steps to ensure accuracy. The process of verifying each step is a crucial aspect of problem-solving in mathematics, ensuring a high degree of confidence in the final answer. We will particularly focus on the calculation of 'c' and its application in finding the foci coordinates.

Recalculating 'c' and Identifying the Foci: A Detailed Approach

Let's revisit the calculation of 'c' and the subsequent determination of the foci coordinates. We have:

$$a^2 = 16^3 = (2^4)^3 = 2^{12}$$

$$b^2 = 30^3 = (2 \cdot 3 \cdot 5)^3 = 2^3 \cdot 3^3 \cdot 5^3$$

Using the formula $c^2 = a^2 + b^2$:

$$c^2 = 2^{12} + 2^3 \cdot 3^3 \cdot 5^3 = 4096 + 27000 = 31096$$

So, $c = \sqrt{31096}$. Instead of approximating, let's try to simplify the radical. We are looking for a perfect square factor of 31096. Since we know $c^2 = a^2 + b^2$, and we are dealing with a hyperbola that might have integer coordinates for its foci in the answer choices, let's go back to the equation $c^2 = a^2 + b^2$ and try to express $c^2$ in a way that we can factor easily.

We have $a^2 = 16^3 = 4096$ and $b^2 = 30^3 = 27000$. Thus,

$$c^2 = 4096 + 27000 = 31096$$

Now, let's express the foci as (14 ± c, 2). We need to find the value of c such that 14 ± c matches one of the x-coordinates in the answer choices (A, B, C, D). The options are:

• A: (-20, 48)
• B: (-16, 44)
• C: (-2, 30)
• D: (14, 14)

The x-coordinates of the foci should be in the form 14 ± c. Let's analyze each option:

• Option A: If 14 + c = 48, then c = 34. If 14 - c = -20, then c = 34. So, c = 34.
• Option B: If 14 + c = 44, then c = 30. If 14 - c = -16, then c = 30. So, c = 30.
• Option C: If 14 + c = 30, then c = 16. If 14 - c = -2, then c = 16. So, c = 16.
• Option D: This option has the same x-coordinate for both foci, which is not possible for a hyperbola.

Now, we need to check which value of c satisfies the equation $c^2 = a^2 + b^2 = 31096$:

• If c = 34, then $c^2 = 34^2 = 1156$, which is not equal to 31096.
• If c = 30, then $c^2 = 30^2 = 900$, which is not equal to 31096.
• If c = 16, then $c^2 = 16^2 = 256$, which is not equal to 31096.

This analysis indicates that there was an error in the interpretation or transcription of the original problem, or in the answer choices provided. None of the options yield a 'c' value that satisfies the equation $c^2 = a^2 + b^2 = 31096$. However, let's revisit the calculations and potential simplifications.

It seems there was a misunderstanding in calculating $c$. Let's correct it:

We have $a^2 = 16^3 = 4096$ and $b^2 = 30^3 = 27000$. Therefore,

$$c^2 = a^2 + b^2 = 4096 + 27000 = 31096$$

So, $c = \sqrt{31096}$. Now, we must find a way to relate this value to the options given. The options suggest a simpler value of 'c' might be expected. Let's look for a perfect square factor in 31096:

$$31096 = 4 \times 7774$$

This doesn't immediately lead to a simple square root. Let's try dividing by 16 (since 16 is a factor of 4096):

$$31096 / 16 = 1943.5$$

This is not an integer, so 16 is not a perfect square factor. Let's try another approach. Since we are looking for integer coordinates for the foci, let's assume the given options are correct and work backward to see which one fits the equation.

Let's consider Option A: (-20, 2) and (48, 2).

The center is (14, 2). The distance from the center to each focus is c. So:

$$c = 48 - 14 = 34$$

Or

$$c = 14 - (-20) = 34$$

So, if option A is correct, c = 34. Then, $c^2 = 34^2 = 1156$. We need to check if this is equal to $a^2 + b^2$:

$$a^2 + b^2 = 4096 + 27000 = 31096$$

Since 1156 ≠ 31096, option A is incorrect.

Let's consider Option B: (-16, 2) and (44, 2).

$$c = 44 - 14 = 30$$

Or

$$c = 14 - (-16) = 30$$

So, c = 30. Then, $c^2 = 30^2 = 900$. We need to check if this is equal to $a^2 + b^2$:

$$a^2 + b^2 = 4096 + 27000 = 31096$$

Since 900 ≠ 31096, option B is incorrect.

Let's consider Option C: (-2, 2) and (30, 2).

$$c = 30 - 14 = 16$$

Or

$$c = 14 - (-2) = 16$$

So, c = 16. Then, $c^2 = 16^2 = 256$. We need to check if this is equal to $a^2 + b^2$:

$$a^2 + b^2 = 4096 + 27000 = 31096$$

Since 256 ≠ 31096, option C is incorrect.

Option D: (14, -32) and (14, 36). This option represents a vertical hyperbola, but the given equation represents a horizontal hyperbola, so option D is incorrect.

Conclusion: Identifying the Correct Foci of the Hyperbola

After a thorough analysis and step-by-step calculations, it is evident that none of the provided options (A, B, C, or D) accurately represent the foci of the hyperbola defined by the equation $\frac{(z-14)^2}{163}-\frac{(y-2)^2}{303}=1$. Our calculations revealed that $c = \sqrt{31096} \approx 176.34$, and the foci should be located at approximately (14 ± 176.34, 2), which does not correspond to any of the given choices. This discrepancy indicates a potential error in the provided answer options or in the initial formulation of the problem. In such cases, it is crucial to double-check all calculations and assumptions to ensure accuracy. If the problem and calculations are verified, the conclusion remains that none of the provided options are correct. It is important to have confidence in the problem-solving process and the mathematical principles applied, even when the expected answer is not readily available among the given choices. This exercise underscores the importance of meticulous calculation and the application of fundamental principles in solving mathematical problems, particularly those involving conic sections like hyperbolas. It also highlights the significance of critical evaluation of given options against derived solutions to ensure accuracy and validity. Therefore, based on our comprehensive analysis, we conclude that none of the options A, B, C, or D correctly identify the foci of the given hyperbola.