Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}+y^{2}-4 x-6 y-23=0\)

Step-by-Step Solution

Answer

The equation doesn't represent any of the listed conic sections (circle, parabola, ellipse, or hyperbola) due to a negative term on the right hand side.

Step 1: Rearrange the equation

First, arrange the equation by grouping the x and y terms together. So it becomes \( (x^{2}-4x)+(y^{2}-6y)=-23 \)

Step 2: Complete the square for x and y terms

Next, to complete the square, add and subtract square of half of the coefficients of x and y to the equation and rearrange it again, which gives \( (x^{2}-4x+4)+(y^{2}-6y+9)=-23+4+9 \) then simplify this equation to \( (x-2)^{2}+(y-3)^{2}= -10 \)

Step 3: Finalize the type

Here, \( (x-2)^{2}+(y-3)^{2}= -10 \) cannot describe any of the conic sections where both the x and y terms are squared. Conic sections are always positive, but here we have a negative term on the right hand side.

Key Concepts

Circle

A circle is a special type of conic section that is defined as the set of all points in a plane that are equidistant from a given point called the center. The equation of a circle in standard form is \[ (x-h)^2 + (y-k)^2 = r^2 \]where

• \((h, k)\) is the center of the circle, and
• \(r\) is the radius.

To identify the circle from an equation, look for equations where the coefficients of \(x^2\) and \(y^2\) are equal and positive. If you want to solve a problem involving circles, make sure that both terms \((x-h)^2\) and \((y-k)^2\) are positive, allowing you to rearrange them to reach this form. If one of the terms is negative or the equation equates to a negative value, it cannot represent a circle due to the nature of geometrical distance, which cannot be negative.

Parabola

A parabola is another conic section characterized by a set of points where each point is equidistant from a fixed point, called the focus, and a fixed line, called the directrix. The equation for a parabola can have different forms depending on its orientation.

• If it opens up or down: \[ y = ax^2 + bx + c \]
• If it opens left or right: \[ x = ay^2 + by + c \]

Parabolas have distinctive properties, such as having only one squared term variable (either \(x^2\) or \(y^2\)), which helps differentiate it from other conic sections. In real-life scenarios, parabolas model many natural phenomena, such as the path of projectiles or reflective properties in satellite dishes.

Ellipse

An ellipse is a conic section that looks like a stretched circle. It's the set of all points for which the sum of the distances to two fixed points (foci) is constant. A circle can be considered a special type of ellipse where these foci coincide.The standard form of an ellipse has two variations:

• Horizontal major axis: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]
• Vertical major axis: \[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]

where

• \((h, k)\) is the center of the ellipse
• \(a\) is the distance from the center to a vertex along the major axis
• \(b\) is the distance from the center to a vertex along the minor axis.

Identifying an ellipse involves checking whether both \(x\) and \(y\) terms are squared, have positive coefficients, and it's equal to a positive number. Ellipses appear in various real-world contexts, such as planet orbits and optical systems.

Hyperbola

A hyperbola is a type of conic section derived from a geometric shape capable of exhibiting two symmetrical open curves. Hyperbolas are the set of points for which the absolute difference of their distances to two fixed points (foci) is constant.The standard form of a hyperbola is:

• For horizontal transverse axis: \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]
• For vertical transverse axis: \[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]

where

• \((h, k)\) is the center of the hyperbola,
• \(a\) and \(b\) are distances that help define the shape.

Important characteristics of hyperbolas include two squared terms with opposite signs. Often found in physics and engineering, such as in the paths of certain celestial bodies or structures that require stability under stress. Recognizing a hyperbola requires observing the sign of the relationship between the squared terms, ensuring the difference implies separation.