Here is a detailed, elaborate, and educational solution to the problem, designed to guide you through each step with clear explanations and insights.

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Key Concept: Equation of a Tangent to a Hyperbola

To solve this problem, we will utilize a fundamental formula for finding the equation of a tangent line to a hyperbola when its slope is known.

For a hyperbola given by its standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ The equation of a tangent line with a known slope $m$ is given by the formula: $y = mx \pm \sqrt{a^2m^2 - b^2}$ This formula is incredibly powerful and is derived using calculus (finding the derivative $\frac{dy}{dx}$ and setting it equal to $m$) or by using discriminant conditions (substituting $y = mx+c$ into the hyperbola equation and setting the discriminant of the resulting quadratic to zero).

Important Conditions:

• For real tangents to exist, the expression under the square root must be non-negative: $a^2m^2 - b^2 \ge 0$.
• If $a^2m^2 - b^2 < 0$, no real tangents with that slope exist.
• If $a^2m^2 - b^2 = 0$, the tangents are actually the asymptotes of the hyperbola.

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Step 1: Standardize the Equation of the Hyperbola

Why this step is necessary: The tangent formula relies on the parameters $a^2$ and $b^2$, which can only be accurately identified when the hyperbola's equation is in its standard form. Without this step, we risk using incorrect values, leading to an erroneous result.

Given Equation: The equation of the hyperbola is $4x^2 - 5y^2 = 20$.

Action: Our goal is to transform this equation into the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The key characteristic of the standard form is that the right-hand side of the equation must be $1$. To achieve this, we divide every term in the given equation by $20$: $\frac{4x^2}{20} - \frac{5y^2}{20} = \frac{20}{20}$

Explanation: Now, we simplify the fractions on the left-hand side: $\frac{x^2}{5} - \frac{y^2}{4} = 1$ By comparing this simplified equation with the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, we can directly identify the values of $a^2$ and $b^2$:

• $a^2 = 5$
• $b^2 = 4$ These values are now correctly identified and ready for substitution into our tangent formula.

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Step 2: Determine the Slope of the Tangent ($m$)

Why this step is necessary: The tangent formula requires the slope, $m$, of the tangent line. The problem provides information about a line parallel to the tangent. The property of parallel lines is what allows us to determine $m$.

Given Information: The problem states that the tangent line is parallel to the line $x - y = 2$.

Action: To find the slope of any straight line, it's easiest to convert its equation into the slope-intercept form, $y = mx + c$, where $m$ directly represents the slope. Let's rearrange the given line equation: $x - y = 2$ To isolate $y$, we can move $y$ to the right side and $2$ to the left side: $x - 2 = y$ Or, written in the standard slope-intercept form: $y = 1x - 2$

Explanation: By comparing $y = 1x - 2$ with the general slope-intercept form $y = mx + c$, we can clearly see that the coefficient of $x$ is $1$. Therefore, the slope of the given line is $m_{line} = 1$.

Since the tangent line is parallel to this line, a fundamental property of parallel lines dictates that they must have the same slope. Thus, the slope of our tangent line, $m_{tangent}$, is also $1$. So, we have $m = 1$.

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Step 3: Apply the Tangent Formula

Why this step is necessary: With the hyperbola parameters ($a^2, b^2$) and the tangent's slope ($m$) now determined, we have all the necessary components to directly use the established formula for the tangent equation.

Identified Parameters:

• From Step 1: $a^2 = 5$
• From Step 1: $b^2 = 4$
• From Step 2: $m = 1$

Action: Substitute these values into the tangent formula $y = mx \pm \sqrt{a^2m^2 - b^2}$: $y = (1)x \pm \sqrt{(5)(1)^2 - 4}$ Perform the calculations inside the square root: $y = x \pm \sqrt{5 \times 1 - 4}$ $y = x \pm \sqrt{5 - 4}$ $y = x \pm \sqrt{1}$ $y = x \pm 1$

Explanation: This step directly provides us with the equations for the two possible tangent lines to the hyperbola $\frac{x^2}{5} - \frac{y^2}{4} = 1$ that have a slope of $1$. The $\pm$ sign indicates that there are generally two such tangents for a hyperbola with a given slope (one for each branch of the hyperbola).

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Step 4: Express in General Form and Compare with Options

Why this step is necessary: The result from the tangent formula gives us the equations in slope-intercept form. To match these with the multiple-choice options, which are typically presented in the general form $Ax + By + C = 0$, we need to rearrange our derived equations.

Derived Tangent Equations:

1. $y = x + 1$
2. $y = x - 1$

Action: Let's convert both equations into the general form $Ax + By + C = 0$:

1. For the equation $y = x + 1$: Subtract $y$ from both sides to get all terms on one side: $0 = x - y + 1$ So, one tangent equation is: $x - y + 1 = 0$

2. For the equation $y = x - 1$: Subtract $y$ from both sides: $0 = x - y - 1$ So, the other tangent equation is: $x - y - 1 = 0$

Explanation: Now we compare our derived tangent equations with the given multiple-choice options: (A) $x - y + 9 = 0$ (B) $x - y - 3 = 0$ (C) $x - y + 1 = 0$ (D) $x - y + 7 = 0$

Our derived equation $x - y + 1 = 0$ perfectly matches option (C).

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Addressing the Discrepancy with the Provided Answer

The problem statement indicates that the "Correct Answer" is (A) $x - y + 9 = 0$. However, based on our meticulous and step-by-step application of the standard mathematical formula for tangents to a hyperbola, and given the specific hyperbola equation and parallel line, the derived tangent equations are $x - y + 1 = 0$ and $x - y - 1 = 0$.

Let's briefly verify why option (A) cannot be correct: If $x - y + 9 = 0$ (which implies $y = x + 9$) were a tangent with slope $m=1$, then according to the formula $y = mx \pm \sqrt{a^2m^2 - b^2}$, we would need $9 = \pm \sqrt{a^2m^2 - b^2}$. This means $9^2 = a^2m^2 - b^2$, so $81 = a^2m^2 - b^2$. However, we calculated $a^2m^2 - b^2 = (5)(1)^2 - 4 = 5 - 4 = 1$. Since $81 \neq 1$, option (A) ($x - y + 9 = 0$) cannot be a tangent to the given hyperbola with the specified slope.

Therefore, based on rigorous mathematical derivation, option (C) is the mathematically correct answer to the question as posed. It is possible that there is an error in the question's options or the designated correct answer. In a competitive exam, you should always trust your derivation.

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Important Tips and Common Mistakes

• Standard Form is Paramount: Always, always begin by converting the conic section's equation to its standard form ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ for hyperbola, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ for ellipse, etc.). Incorrectly identifying $a^2$ and $b^2$ is a very common source of errors. Remember that for a hyperbola of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, $a^2$ is always under the $x^2$ term, and $b^2$ is always under the $y^2$ term (even if $a^2 < b^2$).
• Hyperbola vs. Ellipse Formulas: Be extremely careful with the signs in the tangent formula.
  • For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $y = mx \pm \sqrt{a^2m^2 - b^2}$
  • For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $y = mx \pm \sqrt{a^2m^2 + b^2}$ The difference is subtle but crucial!
• Slope Calculation Accuracy: Double-check your calculation of the slope ($m$) from the given line equation. Remember the properties of parallel lines ($m_1 = m_2$) and perpendicular lines ($m_1 m_2 = -1$).
• Two Tangents: The $\pm$ sign in the formula indicates that there are generally two tangents with a given slope for a hyperbola (unless the radical is zero, in which case they are asymptotes). Both possibilities should be considered and checked against options.
• Trust Your Calculation: If your derived answer doesn't match any of the options (or matches a different option than the given answer key), meticulously recheck your steps. If your calculations remain consistent and correct, it's highly likely there's an error in the question or options. This is a critical skill in competitive exams.

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Summary and Key Takeaway

This problem serves as an excellent example of applying the standard formula for finding the equation of a tangent to a hyperbola when its slope is provided. The solution process is systematic and involves three critical stages:

1. Standardizing the hyperbola equation: This is fundamental for correctly identifying the parameters $a^2$ and $b^2$.
2. Determining the slope ($m$) of the tangent: This is achieved by understanding the relationship between parallel lines.
3. Applying the tangent formula: $y = mx \pm \sqrt{a^2m^2 - b^2}$ is the core formula that directly yields the tangent equations.

By diligently following these steps, we found the tangent equations to be $x - y + 1 = 0$ and $x - y - 1 = 0$. Comparing these with the provided options, $x - y + 1 = 0$ matches option (C). This problem underscores the importance of a deep understanding of conic section formulas and the discipline to apply them accurately, even when faced with potential inconsistencies in question data.