Problem 22
Question
Sketch the graph of the given equation. \(\frac{(x+3)^{2}}{4}+\frac{(y-2)^{2}}{8}=0\)
Step-by-Step Solution
Verified Answer
The graph is a single point at (-3, 2).
1Step 1: Identify the Type of Equation
The given equation is \( \frac{(x+3)^{2}}{4} + \frac{(y-2)^{2}}{8} = 0 \). This equation has the form \( \frac{(x-h)^{2}}{a} + \frac{(y-k)^{2}}{b} = c \), where \( c = 0 \). Normally this represents an ellipse, but since \( c = 0 \), it actually represents a degenerate case.
2Step 2: Understand the Degenerate Case
In an ellipse equation like \( \frac{(x-h)^{2}}{a} + \frac{(y-k)^{2}}{b} = 0 \), both terms must equal zero for the equation to hold. This means \( (x+3)^2 = 0 \) and \( (y-2)^2 = 0 \). This implies specific points rather than a shape.
3Step 3: Solve for the Intersection Point
Solve \( (x+3)^2 = 0 \), which gives \( x = -3 \). Similarly, solve \( (y-2)^2 = 0 \), giving \( y = 2 \). Thus, the only solution that satisfies both conditions is the point \( (-3, 2) \).
4Step 4: Sketch the Graph
Plot the point \( (-3, 2) \) on the coordinate plane. Since this is a degenerate case of an ellipse, there are no lines or curves to draw; the graph consists solely of this single point.
Key Concepts
Degenerate EllipseGraphing EquationsIntersection PointCoordinate Plane
Degenerate Ellipse
A degenerate ellipse is a special case in the family of ellipses. Normally, an ellipse is a curved shape defined by a set of points, all situated a certain distance from two focal points. The standard form of an ellipse equation is given by \( \frac{(x-h)^{2}}{a^2} + \frac{(y-k)^{2}}{b^2} = 1 \).
However, when the constant on the right-hand side is zero, as in \( \frac{(x-h)^{2}}{a} + \frac{(y-k)^{2}}{b} = 0 \), the ellipse doesn't exist as a shape anymore. Instead, it degenerates into a single point.
This happens because each squared term must individually be zero to satisfy the equation. In other words, the points that would make up the ellipse collapse down to a single intersection point.
However, when the constant on the right-hand side is zero, as in \( \frac{(x-h)^{2}}{a} + \frac{(y-k)^{2}}{b} = 0 \), the ellipse doesn't exist as a shape anymore. Instead, it degenerates into a single point.
This happens because each squared term must individually be zero to satisfy the equation. In other words, the points that would make up the ellipse collapse down to a single intersection point.
- This point corresponds to the ellipse's center: found by solving \((x-h)^2 = 0\) and \((y-k)^2 = 0\).
- In our specific equation, solving these gives us the point \((-3, 2)\).
Graphing Equations
Graphing equations involves plotting solutions on a coordinate plane. Typically, this process can create shapes such as lines, parabolas, circles, and ellipses. The distinctive form of each equation helps determine the graph's appearance.
For conic sections, such as ellipses, we use their standard equations to understand what to expect from the graph. A non-zero constant yields a complete curve, but if the constant is zero, as seen in a degenerate ellipse, the only graphable component is a single point.
For conic sections, such as ellipses, we use their standard equations to understand what to expect from the graph. A non-zero constant yields a complete curve, but if the constant is zero, as seen in a degenerate ellipse, the only graphable component is a single point.
- This simplification results from both squared terms needing to verify to zero.
- No extended curve appears, creating a unique instance where only a solitary point is marked on the graph.
Intersection Point
The intersection point is critical when graphing degenerate conic sections like a degenerate ellipse. Essentially, it's the single-point representation of a shape that doesn't actually form.
In the process of solving the equation, we equate each squared term to zero. Solving the identities \((x-h)^2 = 0\) and \((y-k)^2 = 0\) separately, the coordinates of the intersection point can be determined.
In the process of solving the equation, we equate each squared term to zero. Solving the identities \((x-h)^2 = 0\) and \((y-k)^2 = 0\) separately, the coordinates of the intersection point can be determined.
- In our exercise, solving \((x+3)^2=0\) leads to \(x=-3\).
- Similarly, solving \((y-2)^2=0\) results in \(y=2\).
Coordinate Plane
A coordinate plane is a two-dimensional surface where you plot points using a pair of numerical coordinates. These coordinates are taken from two fixed perpendicular lines, known as the x-axis (horizontal) and the y-axis (vertical). The point of intersection of these axes is the origin, denoted by \((0, 0)\).
Understanding the coordinate plane is essential when dealing with graphs of equations. It allows mathematicians to visualize complex equations as simple visual representations. In the case of a degenerate ellipse, only one point is plotted, highlighting its unique behavior.
Understanding the coordinate plane is essential when dealing with graphs of equations. It allows mathematicians to visualize complex equations as simple visual representations. In the case of a degenerate ellipse, only one point is plotted, highlighting its unique behavior.
- The intersection point \((-3, 2)\) lies in the plane, plotted by moving 3 units left of the origin along the x-axis and 2 units up along the y-axis.
- Even when an equation simplifies to a single point, knowing precisely where it should be on the plane is crucial.
Other exercises in this chapter
Problem 21
Find the Cartesian equations of the graphs of the given polar equations. $$ r \sin \theta-1=0 $$
View solution Problem 21
Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$
View solution Problem 22
Sketch the region in the second quadrant that is inside the cardioid \(\quad r=2+2 \sin \theta\) and outside the cardioid \(r=2+2 \cos \theta\), and find its ar
View solution Problem 22
Find the Cartesian equations of the graphs of the given polar equations. $$ r^{2}-6 r \cos \theta-4 r \sin \theta+9=0 $$
View solution