Question

Find the equation of the set of points \(P\) satisfying the given conditions. The difference of the distances of \(P\) from \((\pm 7,0)\) is 12 .

Short answer

The equation is \(\frac{x^2}{36} - \frac{y^2}{13} = 1\).

Step-by-step solution

Understanding the Problem

We are tasked to find the equation of a set of points such that for any point \((x, y)\) in this set, the difference in distances from two fixed points (foci) \((7, 0)\) and \((-7, 0)\) is 12.

Identify the Conic

The condition given—that the absolute difference of the distances to two fixed points is a constant—naturally describes a hyperbola. We can use this information to proceed with deriving the equation.

Use the Hyperbola Definition

For a hyperbola centered at the origin, with foci at \((c, 0)\) and \((-c, 0)\), the relationship is given by \(|d_1 - d_2| = 2a\). Here \(d_1\) and \(d_2\) are the distances from a point \((x, y)\). In our problem, \(c = 7\) and thus \(2a = 12\), so \(a = 6\).

General Hyperbola Equation

The equation of a hyperbola with the transverse axis on the x-axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). We now need to find \(b\) using the relation \(b^2 = c^2 - a^2\).

Calculate \(b^2\)

Substituting the known values of \(c\) and \(a\), we find \(b^2 = 7^2 - 6^2 = 49 - 36 = 13\).

Formulate the Equation

Substitute \(a = 6\) and \(b^2 = 13\) into the standard hyperbola equation to get the final answer: \(\frac{x^2}{36} - \frac{y^2}{13} = 1\).

Key concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. There are four main types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each of these conic sections can be described by a unique type of equation derived from the general quadratic equation in two variables. In this particular exercise, we are dealing with a hyperbola. Hyperbolas are formed when the plane cuts both nappes of the cone. They consist of two disjoint curves called branches, which open in opposite directions. Hyperbolas have interesting properties, including two fixed points called foci that determine the curve's shape. A key feature of hyperbolas is that the difference in distances from any point on the hyperbola to the foci is constant.

Distance Formula

To understand the equation of a hyperbola, it is essential to grasp the concept of distance between points in a plane. The distance formula is a way to calculate the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the coordinate system.The distance formula is expressed as:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In hyperbolas, the formula helps compute the distances from any point \( (x, y) \) on the hyperbola to the foci. By knowing these distances, we can understand and derive the equation of the hyperbola. Whenever you hear "difference of distances," remember that hyperbolas are involved, and the distance formula will be a valuable tool.

Transverse Axis

The transverse axis is a central concept when studying hyperbolas. It is the line segment that passes through the two foci and the center of the hyperbola. Importantly, the transverse axis is also the major axis of the hyperbola and lies along its line of symmetry. For a hyperbola with a horizontal transverse axis, such as in our exercise, the axis aligns with the x-axis. The length of the transverse axis is \(2a\), where \(a\) is the semi-major axis length derived from the hyperbola's defining property. Here, the foci are \( (7, 0) \) and \((-7, 0)\), and the transverse axis's length is 12, corresponding to \(2a = 12\), so \(a = 6\). Along the transverse axis, every point's x-coordinate is essential for writing the standard equation of the hyperbola.

Foci of Hyperbola

The foci (singular: focus) of a hyperbola are two crucial fixed points that play a central role in defining its geometry. The defining rule of a hyperbola is that the absolute difference in distances from any point on the hyperbola to these two foci is constant.In our exercise, the foci are strategically placed at \( (7, 0)\) and \((-7, 0)\). These coordinates tell us that the hyperbola is symmetrically centered at the origin. The constant difference of distances, equaling 12 in this case, derived the hyperbola equation.The relationship between the foci and the equation of the hyperbola involves the parameters \( a \) and \( b \), where \(b^2 = c^2 - a^2\) and \(c\) is the distance from the center to a focus. This formula helps establish \(b\) which completes the standard equation of the hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Understanding these elements contributes to a deeper understanding of the hyperbola's nature.