1. Understanding the Fundamentals of Hyperbolas

Before we dive into solving the problem, let's establish a strong foundation by reviewing the essential concepts and formulas related to hyperbolas. This will ensure every step in our solution is clear and well-justified.

• Standard Equation of a Hyperbola (Horizontal Transverse Axis): A hyperbola centered at the origin, with its foci lying on the x-axis (meaning its transverse axis is horizontal), has the standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here, $a$ represents the length of the semi-transverse axis (half the distance between the vertices), and $b$ represents the length of the semi-conjugate axis.

• Coordinates of Foci: For such a hyperbola, the coordinates of its foci are given by $(\pm ae, 0)$, where $e$ is the eccentricity of the hyperbola. For any hyperbola, the eccentricity $e$ must be greater than 1 ($e > 1$).

• Relationship between $a, b,$ and $e$: The parameters $a, b,$ and $e$ are intrinsically linked by the equation: $b^2 = a^2(e^2 - 1)$ This can be rearranged into a very useful form: $a^2e^2 = a^2 + b^2$. This particular form is often more convenient when the product $ae$ (the distance from the center to a focus) is known.

• Equation of the Tangent to a Hyperbola at a Point: If a point $P(x_1, y_1)$ lies on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the equation of the tangent line to the hyperbola at this specific point is given by: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ This is a standard formula derived from differential calculus or by using the 'T=0' substitution method.

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2. Step-by-Step Solution

Let's apply these concepts to systematically solve the given problem.

Step 1: Determine Hyperbola Parameters from the Foci

• Given Information: We are told that the foci of the hyperbola are located at $(\pm 2, 0)$.
• Applying the Formula: We recall that for a standard hyperbola with a horizontal transverse axis, the foci are at $(\pm ae, 0)$.
• Deduction of $ae$: By comparing the given foci with the standard form, we directly find the value of $ae$: $ae = 2$
• Connecting $ae$ to $a^2$ and $b^2$: We use the fundamental relationship $a^2e^2 = a^2 + b^2$.
  • Why this form? Since we know $ae$, squaring it directly gives us $(ae)^2 = a^2e^2$, which simplifies the substitution.
• Substitution: Squaring $ae=2$, we get $(ae)^2 = 2^2 = 4$. Substituting this into the relationship: $a^2 + b^2 = 4 \quad \ldots(1)$ This is our first crucial equation, linking the unknown parameters $a^2$ and $b^2$ that define our hyperbola.

Step 2: Use the Given Point to Form a Second Equation

• Given Information: The hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$.
• Principle: If a point lies on a curve, its coordinates must satisfy the equation of that curve.
• Applying the Hyperbola Equation: We substitute $x = \sqrt{2}$ and $y = \sqrt{3}$ into the standard equation of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $\frac{(\sqrt{2})^2}{a^2} - \frac{(\sqrt{3})^2}{b^2} = 1$
• Simplifying the Equation: $\frac{2}{a^2} - \frac{3}{b^2} = 1 \quad \ldots(2)$ This is our second equation, providing another relationship between $a^2$ and $b^2$.

Step 3: Solve the System of Equations for $a^2$ and $b^2$

• Objective: We now have a system of two equations with two unknowns ($a^2$ and $b^2$). Our goal is to solve this system to find the specific values that define the hyperbola.
  1. $a^2 + b^2 = 4$
  2. $\frac{2}{a^2} - \frac{3}{b^2} = 1$
• Strategy: From equation (1), it's straightforward to express $a^2$ in terms of $b^2$: $a^2 = 4 - b^2$
• Substitution: Substitute this expression for $a^2$ into equation (2): $\frac{2}{(4 - b^2)} - \frac{3}{b^2} = 1$
• Combine and Simplify (Algebraic Steps): To solve for $b^2$, we combine the fractions on the left-hand side by finding a common denominator: $\frac{2b^2 - 3(4 - b^2)}{(4 - b^2)b^2} = 1$ Now, clear the denominator by multiplying both sides by $(4 - b^2)b^2$: $2b^2 - 12 + 3b^2 = (4 - b^2)b^2$ $5b^2 - 12 = 4b^2 - b^4$
• Rearrange into a Quadratic Equation (in terms of $b^2$): Move all terms to one side to form a standard quadratic equation. It's often easier to work with a positive leading coefficient: $b^4 + b^2 - 12 = 0$
• Factorize the Quadratic: This is a quadratic equation in $b^2$. Let $X = b^2$. Then the equation becomes $X^2 + X - 12 = 0$. We look for two numbers that multiply to -12 and add to 1 (which are 4 and -3). $(b^2 + 4)(b^2 - 3) = 0$
• Solve for $b^2$: This factorization gives us two potential solutions for $b^2$: $b^2 = 3 \quad \text{or} \quad b^2 = -4$
• Reject Invalid Solution:
  • Important Tip: By definition, $b^2$ represents the square of a real length (the semi-conjugate axis length), and thus it must always be a positive value.
  • Therefore, we must reject $b^2 = -4$ as it is physically impossible for a real hyperbola. $b^2 = 3$
• Find $a^2$: Substitute the valid value of $b^2 = 3$ back into our expression for $a^2$ from equation (1): $a^2 = 4 - b^2 = 4 - 3 = 1$
• Verification: Both $a^2 = 1$ and $b^2 = 3$ are positive, confirming that a real hyperbola exists with these parameters. If either value had been zero or negative, it would indicate an error in calculation or an impossible scenario for a real hyperbola.

Step 4: Write the Equation of the Hyperbola

• Using the derived values: Now that we have $a^2 = 1$ and $b^2 = 3$, we can write down the specific equation of the hyperbola that meets all the problem's conditions: $\frac{x^2}{1} - \frac{y^2}{3} = 1$

Step 5: Find the Equation of the Tangent at Point P

• Given Point of Tangency: The problem specifies that the tangent is at point $P(x_1, y_1) = (\sqrt{2}, \sqrt{3})$.
• Applying the Tangent Formula: We use the standard formula for the tangent to a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at a point $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$
• Substitution: Substitute the values $x_1 = \sqrt{2}$, $y_1 = \sqrt{3}$, $a^2 = 1$, and $b^2 = 3$ into the formula: $\frac{x(\sqrt{2})}{1} - \frac{y(\sqrt{3})}{3} = 1$
• Simplified Tangent Equation: $\sqrt{2}x - \frac{\sqrt{3}}{3}y = 1$ This is the equation of the tangent line to the hyperbola at point $P(\sqrt{2}, \sqrt{3})$.

Step 6: Check Which Option Satisfies the Tangent Equation

• Objective: We need to find which of the given points lies on the tangent line $\sqrt{2}x - \frac{\sqrt{3}}{3}y = 1$. We do this by substituting the coordinates of each option into the tangent equation and checking if the equation holds true.

• Option (A): $(2\sqrt{2}, 3\sqrt{3})$ Substitute $x = 2\sqrt{2}$ and $y = 3\sqrt{3}$ into the left-hand side (LHS) of the tangent equation: $\text{LHS} = \sqrt{2}(2\sqrt{2}) - \frac{\sqrt{3}}{3}(3\sqrt{3})$ Let's simplify this step-by-step: $\text{LHS} = (2 \times \sqrt{2} \times \sqrt{2}) - \left( \frac{3\sqrt{3} \times \sqrt{3}}{3} \right)$ $\text{LHS} = (2 \times 2) - \left( \frac{3 \times 3}{3} \right)$ $\text{LHS} = 4 - 3$ $\text{LHS} = 1$ Since the LHS equals 1, which is the RHS of the tangent equation, the point $(2\sqrt{2}, 3\sqrt{3})$ lies on the tangent line.

• Conclusion: Option (A) is the correct answer. In a multiple-choice examination, once the correct option is found, there is no need to check the remaining options, saving valuable time.

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3. Summary and Key Takeaway

This problem is an excellent exercise that integrates multiple key concepts related to hyperbolas. The methodical approach involved:

1. Utilizing Foci Information: Extracting the value of $ae$ from the given foci and then using the relationship $a^2e^2 = a^2 + b^2$ to form the first equation ($a^2 + b^2 = 4$). This is a common strategy when foci coordinates are given.
2. Employing the Point on the Hyperbola: Substituting the coordinates of the given point $P(\sqrt{2}, \sqrt{3})$ into the hyperbola's standard equation to generate a second equation relating $a^2$ and $b^2$ ($\frac{2}{a^2} - \frac{3}{b^2} = 1$).
3. Solving the System: Solving the two simultaneous equations for $a^2$ and $b^2$. Remember to always verify that $a^2$ and $b^2$ are positive, as they represent squares of real lengths.
4. Forming the Tangent Equation: Using the derived values of $a^2$ and $b^2$ along with the point of tangency $P(\sqrt{2}, \sqrt{3})$ to write the equation of the tangent line.
5. Verifying Options: Checking each option by substituting its coordinates into the tangent equation.

This problem highlights the importance of mastering the fundamental definitions and formulas of conic sections and applying them systematically to solve complex problems.