Problem 28 - Calculus Single Variable

Question

The center of an ellipse or hyperbola is the point of intersection of its axes of symmetry. Each state whether the graph of the given Cartesian equation is an ellipse or hyperbola. Determine its standard form and center. \(-(x+5)^{2}+y^{2}=9\)

Step-by-Step Solution

Answer

The graph is a hyperbola with center at (-5, 0) and its standard form is \(\frac{y^2}{9} - \frac{(x+5)^2}{9} = 1\).

Step 1: Identify the Type of Conic

The equation given is \[-(x+5)^{2}+y^{2}=9\]. Notice the negative sign in front of the \((x+5)^2\) term, which indicates that the conic is centered on the y-axis rather than the x-axis and features a subtraction in its equation. Therefore, it is a hyperbola, not an ellipse.

Step 2: Rewrite in Standard Form

The standard form of a hyperbola with a vertical transverse axis is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\].We need to rewrite the provided equation in this form: divide through by 9 to get:\[\frac{y^2}{9} - \frac{(x+5)^2}{9} = 1\].

Step 3: Identify the Center

From the rewritten standard form \[\frac{y^2}{9} - \frac{(x+5)^2}{9} = 1\], we can identify the center of the hyperbola. The center is at \((h, k)\) which translates from our equation to \((-5, 0)\) given that we have \((x+5)^{2}\) and \(y^2\).

Key Concepts

Hyperbola

A hyperbola is a fascinating conic section that consists of two distinct curves, often resembling mirror images of each other. This shape is created when a plane intersects a double cone at a particular angle. Unlike ellipses, where the curves are closed, hyperbolas form open curves extending indefinitely in two directions.
When studying hyperbolas, it's essential to understand that they have two axes of symmetry—the transverse axis and the conjugate axis. These axes are crucial in determining the orientation and properties of the hyperbola.

• Transverse Axis: This is the line through the vertices of the hyperbola, essentially the axis along which the hyperbola opens. It can be either horizontal or vertical.
• Conjugate Axis: Perpendicular to the transverse axis, this axis is through the points known as co-vertices. It doesn't intersect with the hyperbola but plays a role in its geometric definition.

Knowing these axes helps identify the hyperbola's orientation and relates directly to the sign and position of terms in its equation.
In distinguishing hyperbolas from other conic sections such as ellipses, the sign in front of the squared terms in their equations is key. The presence of a subtractive term, as seen with the negative sign in this exercise's equation \[-(x+5)^{2}+y^{2}=9\], indicates it's a hyperbola.

Cartesian Equation

The Cartesian equation is the mathematical representation of a curve or surface in two-dimensional or three-dimensional space. For a hyperbola, the Cartesian equation often reveals the relationship between the variables involved in the standard coordinate planes.
In the simple case of a two-dimensional plane, the general form of a hyperbola's equation can be identified by the difference between the squares of the variables. For example:

• If the form is \[(x-h)^2/a^2 - (y-k)^2/b^2 = 1\], the hyperbola is horizontal, opening left and right.
• If the form is \[(y-k)^2/a^2 - (x-h)^2/b^2 = 1\], the hyperbola is vertical, opening upward and downward.

Identifying these forms from a Cartesian equation, like \[-(x+5)^{2}+y^{2}=9\], allows us to visualize and understand the curve's properties. After some rearrangement, this particular equation fits the vertical hyperbola form,
indicating a specific orientation and center of the hyperbola at point \((-5, 0)\).
Thus, the Cartesian equation is indispensable for transforming and verifying the characteristics of hyperbolas and other conic sections.

Standard Form of Conics

The standard form of conic sections is a way to write equations that reveals their essential properties, such as position, orientation, and size. Understanding the standard form allows for easy identification of the conic section's type—whether it be an ellipse, parabola, or, in our focus, a hyperbola.
For hyperbolas, the standard form helps to pinpoint:

• The Center: Given by \((h, k)\) in the hyperbola's equation, the center is where the axes intersect.
• Axes: The constants \(a^2\) and \(b^2\) define the lengths and orientation of the transverse and conjugate axes.

In our exercise's rewritten form \[(y-k)^2/a^2 - (x-h)^2/b^2 = 1\], converting \[-(x+5)^{2}+y^{2}=9\] into \[ \frac{y^2}{9} - \frac{(x+5)^2}{9} = 1 \] exposes the center at \((-5, 0)\), confirming the orientation of the vertical transverse axis.
Interpreting the equation in this form clarifies not just the position and dimensions of the hyperbola,
but also its geometric behavior, aiding in deeper comprehension and graphical representation of the curve.