Question

Find the equation of the set of points \(P\) satisfying the given conditions. The sum of the distances of \(P\) from \((\pm 4,0)\) is 14 .

Short answer

The equation of the ellipse is \( (x^2/49) + (y^2/33) = 1 \).

Step-by-step solution

Recognize the Geometric Figure

The problem describes a set of points such that the sum of the distances from two fixed points, \( (4,0) \) and \( (-4,0) \), is constant (14). This definition corresponds to an ellipse.

Use the Standard Form of an Ellipse

The standard form of an ellipse centered at the origin with foci at \( ( ext{-}c,0) \) and \( (c,0) \) is \((x^2/a^2) + (y^2/b^2) = 1\), where \(2a\) is the sum of the distances to the foci. Here, \(2a = 14\), so \(a = 7\).

Identify the Value of c

The distance between the foci \( (4,0) \) and \( (-4,0) \) is \(2c = 8\), thus \(c = 4\).

Find b Using the Relationship in the Ellipse

For an ellipse, the relationship \(c^2 = a^2 - b^2\) holds. We know \(c = 4\) and \(a = 7\). Substitute these to find \((4)^2 = 7^2 - b^2\).

Solve for b

We substitute \(c = 4\) and \(a = 7\) into the equation \(c^2 = a^2 - b^2\): \(16 = 49 - b^2\). Solve for \(b^2\): \(b^2 = 49 - 16 = 33\). Thus, \(b = \sqrt{33}\).

Write the Equation of the Ellipse

Substitute \(a = 7\), \(b = \sqrt{33}\) into the standard form equation of an ellipse, giving us \((x^2/49) + (y^2/33) = 1\).

Key concepts

Distance Formula

Understanding how to calculate the distance between two points is crucial when learning about ellipses. The **distance formula** is derived from the Pythagorean theorem and is used to find the length of the line segment between two points in a plane. The formula itself is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

If we think about how this formula applies to ellipses, it helps us understand why the sum of distances from a point on an ellipse to each focus is constant. This geometric property is a key characteristic of ellipses. When solving problems involving ellipses, this foundational knowledge of distance calculations becomes very useful.
The problem often involves finding the sum of distances from various points on the ellipse to its foci, which demands familiarity with this basic geometric principle.

Foci of an Ellipse

The **foci** (singular: focus) are two fixed points on the interior of an ellipse. The defining property of an ellipse is that for any point on the ellipse, the sum of the distances to the foci is constant. This is what differentiates an ellipse from a circle, which has only one center.

Understanding how to determine the foci is essential. If an ellipse is centered at the origin and has its foci along the x-axis, the coordinates of the foci will typically be \((\pm c, 0)\), where \(c\) represents the focal distance from the center of the ellipse to each focus.
In algebraic terms, the position of the foci connects to both the semi-major axis \(a\) and the semi-minor axis \(b\) through the relationship \(c^2 = a^2 - b^2\). By this formula, once you know \(a\) and \(b\), calculating \(c\) becomes straightforward and essential for writing the equation of the ellipse.

Standard Form of Ellipse

The **standard form of an ellipse** is an important equation in algebraic geometry. When an ellipse is centered at the origin, its equation in standard form is given by \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where \(a\) is the semi-major axis and \(b\) is the semi-minor axis of the ellipse. These values determine the shape and orientation of the ellipse.

In our problem, we determine \(a\) as half of the constant sum of distances to the foci, known as the major axis length \(2a\). From the given conditions, \(2a = 14\), thus \(a = 7\).
To complete the standard equation, \(b\) must be calculated next using the relationship with the focus distance \(c\). Once \(a\) and \(b\) are known, the equation provides a precise description of all points on the ellipse, defining its scope and extent.

Algebraic Geometry

**Algebraic geometry** is a fascinating field that studies solutions of polynomial equations using an algebraic framework while also including geometric insights. In the case of ellipses and other conic sections, algebraic geometry provides powerful tools to describe and solve geometric problems using equations. This helps visualize shapes and explore their properties in a mathematical form.

An ellipse is a central topic within algebraic geometry due to its complex yet systematic nature. Its characteristics encompass intersections of planes with cones, and through its equations, we delve into solving problems about locus of points.
Using algebraic geometry techniques, we relate terms like the foci distance \(c\), major and minor axes \(a\) and \(b\), constructing comprehensive models that guide us to understand these geometric structures. This union of algebra and geometry not only illustrates shapes but resolves complex real-world problems.