91Ó°ÊÓ

When we talk about conic sections, we are referring to the curves obtained by intersecting a cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas. Each one is defined by the angle at which the plane intersects the cone.

A hyperbola, the focus of our discussion, results from an intersecting plane that cuts through both halves of the cone but not at the angle shallow enough for an ellipse or deeply enough for a parabola. Visually, a hyperbola is made up of two symmetrical open curves, referred to as branches, which mirror each other and extend to infinity. The hyperbola has many properties that are used in various areas of mathematics and the sciences, particularly in the study of orbits and asymptotes.

The hyperbola equation is of the standard form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) for a horizontal hyperbola, or \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) for a vertical hyperbola. Here, \((h,k)\) is the center of the hyperbola, \((a)\) is the distance from the center to the vertices along the transverse axis, and \((b)\) is the distance from the center to the co-vertices along the conjugate axis.

In our exercise, \((a^{2}=9)\) and \((b^{2}=1)\), indicating that the hyperbola is horizontal because the larger number is under \((x^{2})\), which gives the horizontal direction. It's essential for students to identify the form of the equation correctly, as this will determine the orientation and ultimately how to find the foci of the hyperbola.

To find the foci of a hyperbola, we must compute their coordinates by using the formula \((c=\sqrt{a^{2}+b^{2}})\), where \(c\) is the distance from the center of the hyperbola to each focus. The foci always lie on the transverse axis, which is the axis where the \(a\) value lies.

In our given problem, the students are instructed to locate the foci of the hyperbola based on its equation. After substituting the known values from the equation into the foci formula, they find that \(c=\sqrt{10}\). Since we have a horizontal hyperbola, the coordinates of the foci will be at \((\pm c, 0)\), resulting in the points \((\pm \sqrt{10}, 0)\). The students should note that foci are symmetrically located with respect to the center of the hyperbola, and these points are crucial in defining the shape's geometry and determining its asymptotes.