Key Concepts and Formulas

• Standard Form of a Hyperbola: The equation of a hyperbola centered at the origin is given by $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ or $\frac{y^2}{A^2} - \frac{x^2}{B^2} = 1$.
• Tangency Condition for a Hyperbola: The line $y = mx + c$ is tangent to the hyperbola $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ if and only if $c^2 = A^2m^2 - B^2$.

Step-by-Step Solution

Step 1: Convert the given hyperbola equation to its standard form.

The given hyperbola equation is $a^2x^2 - y^2 = b^2$. To apply the tangency condition, we need to express this in the standard form $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$. Divide the equation by $b^2$: $\frac{a^2x^2}{b^2} - \frac{y^2}{b^2} = 1$ $\frac{x^2}{b^2/a^2} - \frac{y^2}{b^2} = 1$ Thus, we have $A^2 = \frac{b^2}{a^2}$ and $B^2 = b^2$. This step is necessary to correctly identify the parameters for the tangency condition.

Step 2: Rewrite the given tangent line equation in slope-intercept form.

The given tangent line equation is $\lambda x - 2y = \mu$. We need to rewrite this in the form $y = mx + c$. Isolate $y$: $-2y = -\lambda x + \mu$ Divide by -2: $y = \frac{\lambda}{2}x - \frac{\mu}{2}$ This transformation allows us to identify the slope and y-intercept of the tangent line.

Step 3: Identify the slope ($m$) and y-intercept ($c$) of the tangent line.

Comparing $y = \frac{\lambda}{2}x - \frac{\mu}{2}$ with the standard form $y = mx + c$, we identify: $m = \frac{\lambda}{2}$ $c = -\frac{\mu}{2}$ These values are crucial for substituting into the tangency condition.

Step 4: Apply the tangency condition.

Substitute the values of $A^2$, $B^2$, $m$, and $c$ into the tangency condition $c^2 = A^2m^2 - B^2$: $\left(-\frac{\mu}{2}\right)^2 = \left(\frac{b^2}{a^2}\right)\left(\frac{\lambda}{2}\right)^2 - b^2$ This step uses the previously identified parameters and the tangency condition to create an equation relating $\lambda$, $\mu$, $a$, and $b$.

Step 5: Simplify the equation to find the desired expression.

Simplify the equation: $\frac{\mu^2}{4} = \frac{b^2}{a^2} \cdot \frac{\lambda^2}{4} - b^2$ Multiply the entire equation by 4: $\mu^2 = \frac{b^2}{a^2}\lambda^2 - 4b^2$ Rearrange the terms to match the form $\left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2$: $4b^2 = \frac{b^2}{a^2}\lambda^2 - \mu^2$ Divide the entire equation by $b^2$: $4 = \frac{\lambda^2}{a^2} - \frac{\mu^2}{b^2}$ Therefore, $\left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2 = 4$ This final simplification isolates the desired expression and provides the numerical answer.

Common Mistakes & Tips

• Be careful with signs when applying the tangency condition. For the hyperbola $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$, the correct condition is $c^2 = A^2m^2 - B^2$.
• Ensure the hyperbola is in standard form before identifying $A^2$ and $B^2$.
• Double-check algebraic manipulations, especially when squaring fractions and multiplying by constants.

Summary

This problem involves finding the relationship between $\lambda$, $\mu$, $a$, and $b$ given that $\lambda x - 2y = \mu$ is tangent to the hyperbola $a^2x^2 - y^2 = b^2$. By converting the hyperbola and line equations to standard forms, applying the tangency condition, and simplifying, we find that $\left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2 = 4$.

The final answer is \boxed{4}, which corresponds to option (D).