91Ó°ÊÓ

Implicit differentiation is a technique used to find the derivative of a function when it is not expressed explicitly as \( y = f(x) \). In the context of our problem, this method is essential because the hyperbola is given as an implicit equation: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]This equation cannot be easily solved for \( y \) in terms of \( x \).
To differentiate implicitly, we take the derivative of each term with respect to \( x \), treating \( y \) as an implicit function that depends on \( x \). This means whenever we differentiate a term involving \( y \), we must apply the chain rule and multiply by \( \frac{dy}{dx} \).
For example, differentiating the left-hand side of the hyperbola equation, we use the product rule and chain rule:

• \( \frac{d}{dx}\left(\frac{x^2}{a^2}\right) = \frac{2x}{a^2} \)
• \( \frac{d}{dx}\left(-\frac{y^2}{b^2}\right) = -\frac{2y}{b^2} \frac{dy}{dx} \)

Setting the derivative of the left side equal to zero provides an equation we can solve for \( \frac{dy}{dx} \), which represents the slope of the tangent to the hyperbola.

The slope of a tangent line to a curve at a specific point describes how steep the line is at that point. For the hyperbola, after performing implicit differentiation, we find the slope \( m \) by solving for \( \frac{dy}{dx} \).
The formula we obtain is:\[ \frac{dy}{dx} = \frac{b^2}{a^2} \frac{x}{y} \]
This expression depends on both \( x \) and \( y \). To find the slope at a particular point \( (x_0, y_0) \), we substitute these values into the equation:\[ m = \frac{b^2}{a^2} \frac{x_0}{y_0} \]

• \( x_0 \) and \( y_0 \) are the coordinates of the point lying on the hyperbola.
• The slope \( m \) provides the steepness of the tangent line at \( (x_0, y_0) \).

This slope is crucial for forming the equation of the tangent line.

The standard form of a hyperbola centered at the origin with the x-axis as its transverse axis is an important concept. It helps us understand the geometric properties of the hyperbola. The form is:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Here:

• \( a \) and \( b \) are the real numbers that determine the shape and orientation of the hyperbola.
• The equation describes a hyperbola with its transverse axis along the x-axis.
• The center is at the origin \( (0, 0) \).

Understanding this form is essential because it allows us to see the symmetry and structure inherent in hyperbolas. This form is also crucial when applying implicit differentiation or finding the tangent lines.
The point \( (x_0, y_0) \) being on the hyperbola means it satisfies this standard equation, a critical part of finding the tangent line.

The tangent line equation at a point on a curve gives the precise linear approximation to the curve at that point. Starting with the slope, after determining \( m \), the equation of the tangent line in point-slope form is:\[ y - y_0 = m(x - x_0) \]
Substituting the slope \( m = \frac{b^2}{a^2} \frac{x_0}{y_0} \), we get:\[ y - y_0 = \frac{b^2}{a^2} \frac{x_0}{y_0} (x - x_0) \]
To match the form \( \frac{x x_0}{a^2} - \frac{y y_0}{b^2} = 1 \), we need to rearrange the equation. This involves simplifying and rearranging terms:

• Expand and simplify the initial equation.
• Multiply through by \( y_0 \) to eliminate the fraction.
• Rewriting all terms to align with the desired form.

Achieving the desired form means confirming the equation represents the exact tangent line required, finalizing the solution's integrity and aligning with the step-by-step process.