Problem 41 - Calculus Early Transcendentals

Question

Find an equation for the conic that satisfies the given conditions. Ellipse, center \((-1,4), \quad\) vertex \((-1,0),\) focus \((-1,6)\)

Step-by-Step Solution

Answer

The equation is \( \frac{(x+1)^2}{16} + \frac{(y-4)^2}{12} = 1 \).

Step 1: Understand the Ellipse Structure

An ellipse has a general equation in standard form as \( (x-h)^2/a^2 + (y-k)^2/b^2 = 1 \) if it's vertically stretched around the center \((h, k)\), where \( a \) is the distance from the center to the vertex along the major axis, and \( b \) is the distance from the center to the vertices along the minor axis. In this case, the center is \((-1, 4)\).

Step 2: Determine the Distance a

Since the center is at \((-1, 4)\) and a vertex at \((-1, 0)\), the distance \(a\) should be calculated vertically to the vertex. Hence, \(a = |4 - 0| = 4\).

Step 3: Compute the Distance c

The focus is given at \((-1, 6)\), and the center is \((-1, 4)\), so the distance from the center to the focus \(c\) is \(c = |6 - 4| = 2\).

Step 4: Use Ellipse Relationship to Find b

The relationship between the distances in an ellipse is \(c^2 = a^2 - b^2\). We know \(a = 4\) and \(c = 2\). Therefore, substitute and solve: \((2)^2 = (4)^2 - b^2\). So, \(4 = 16 - b^2 \), leading to \(b^2 = 12\).

Step 5: Write the Ellipse Equation

Now that we have \( (x-h)^2/a^2 + (y-k)^2/b^2 = 1 \) with \(h = -1\), \(k = 4\), \(a = 4\), and \(b = \sqrt{12}\), substitute these into the equation to get: \[ \frac{(x+1)^2}{4^2} + \frac{(y-4)^2}{12} = 1 \]. This simplifies to \[ \frac{(x+1)^2}{16} + \frac{(y-4)^2}{12} = 1 \].

Key Concepts

Ellipse Geometry

An ellipse is a fascinating geometric shape, appearing like a stretched circle. Unlike a circle, an ellipse has two distinct axes: the major and minor axes. The major axis is the longest diameter of the ellipse, running through its center. The minor axis is perpendicular to the major axis at the center and is the shortest diameter. This arrangement gives the ellipse its oval shape.

In the standard equation of an ellipse, \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]\(h\) and \(k\) are the coordinates of the center, \(a\) represents the semi-major axis, and \(b\) represents the semi-minor axis.

• When the equation has \(a^2 > b^2\), the ellipse is wider horizontally.
• When \(b^2 > a^2\), the ellipse is taller vertically.

The beauty of the ellipse lies in its symmetry and the precise relationship between these axes, allowing it to maintain a smooth, rounded appearance.

Center of Ellipse

The center of an ellipse is like its heart. It is the point from which the ellipse stretches equally in all directions. Knowing the center is crucial for writing the equation and understanding the ellipse's orientation.

For the given exercise, the center is \((-1, 4)\). This means the ellipse is symmetrically distributed around this point. When visualizing, place the center in a coordinate system and draw the axes based on its location.

• The center can often be identified as the midpoint of the line segment connecting the vertices or the foci.
• In designing or analyzing ellipses, the center helps in balancing the layout and ensuring symmetrical properties are upheld.

Overall, the center provides a reference point in the ellipse’s geometry, influencing its positioning and structure.

Conic Sections

Ellipses belong to a special family of curves known as conic sections. They are formed by the intersection of a plane and a double-napped cone. Depending on the angle and position of the plane, different shapes emerge: circles, ellipses, parabolas, or hyperbolas.

An ellipse occurs when the plane cuts the cone at an angle to the base, but not perpendicular. This creates a closed curve. This property of the ellipse being part of conic sections is important in many fields like astronomy and physics.

• Simpler calculations and predictions in elliptical paths can be derived from their conic properties.
• In architecture, the concept supports the creation of dome shapes mimicking celestial movement listed as part of conic beauty.

Thus, understanding ellipses within conic sections broadens the appreciation and application of geometric principles in multiple domains.

Properties of Ellipse

Ellipses possess many interesting properties that make them unique among geometric shapes. One important feature is the concept of foci. In an ellipse, there are two focal points. The sum of the distances from any point on the ellipse to both foci remains constant – a defining property of ellipses.

In our exercise, one focus is \((-1, 6)\). This plays a critical role in determining the shape and equation of the ellipse, as seen in the solution steps.

Moreover, other properties include:

• The eccentricity of an ellipse, ranging from 0 to 1, measuring its deviation from being circular.
• Ellipses reflect the path of celestial objects and are used in designing optical systems due to their reflective properties.

These properties not only define how an ellipse appears but also dictate its physical and real-world applications, from planetary motion to architectural aesthetics.