91Ó°ÊÓ

A hyperbola is a type of smooth curve lying in a plane, defined by its mirror-symmetric property and distance difference from two fixed points, known as foci. The standard equation of a hyperbola centered at point \( (h, k) \) with horizontal transverse axis is given by
\[ \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \]
where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. For a vertical transverse axis, the \( a^{2} \) and \( b^{2} \) terms are switched. Understanding this equation is crucial as it provides the basis for solving problems related to hyperbolas, including finding tangent lines.

The tangency point on a hyperbola is where a line touches the curve without crossing it. This point is significant because at this juncture, the slope of the tangent line is equal to the derivative of the hyperbola at that point. For a hyperbola defined by the equation
\[ \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \]
a point \( (x_1, y_1) \) that lies on the hyperbola will be the tangency point if the line tangent to the hyperbola at \( (x_1, y_1) \) intersects the hyperbola only once. Mathematically, the tangent line will share only that single common point with the hyperbola, which aligns with the condition provided in the exercise problem.

Conic sections are the curves obtained as the intersection of the surface of a cone with a plane. These curves depend on the angle at which the plane cuts the cone and include circles, ellipses, parabolas, and hyperbolas. Conic sections have a rich mathematical history and show up in a variety of natural phenomena, from planetary orbits to the shapes of satellite dishes. A hyperbola specifically is formed when the plane cuts both halves (nappes) of the cone at an angle to its axis. Understanding the properties of conic sections aids in comprehending more advanced topics in mathematics and physics.

The standard form of a hyperbola is an equation that provides a guideline on how all hyperbolas can be represented systematically. For horizontal hyperbolas, this form is
\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]
and for vertical hyperbolas, it is
\[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]
In both cases, the center of the hyperbola is at the point \( (h, k) \) and the denominators \( a^2 \) and \( b^2 \) indicate the squares of the distances from the center to the vertices and co-vertices, respectively. This form is vital for analyzing the hyperbola's geometric properties and for solving problems related to hyperbolas, including the computation of tangent lines at particular points.