Problem 54
Question
We know that \(x^{2}+y^{2}=25\) is the equation of a circle. Rewrite the equation so that the right side is equal to \(1 .\) Which type of conic section does this equation form resemble? In fact, the circle is a special case of this type of conic section. Describe the conditions under which this type of conic section is a circle.
Step-by-Step Solution
Verified Answer
The rewritten equation is \(\frac{x^2}{25} + \frac{y^2}{25} = 1\), resembling an ellipse. The circle is an ellipse where the semi-major and semi-minor axes are equal.
1Step 1: Understand the Equation of a Circle
The given equation, \(x^2 + y^2 = 25\), represents a circle centered at the origin (0,0) with a radius of 5. This is in the standard form of a circle equation \(x^2 + y^2 = r^2\).
2Step 2: Rewrite the Equation
To rewrite the equation so that the right side equals 1, divide each term by 25: \[ \frac{x^2}{25} + \frac{y^2}{25} = 1 \].
3Step 3: Recognize the Form of a Conic Section
The rewritten equation \(\frac{x^2}{25} + \frac{y^2}{25} = 1\) resembles the standard form of an ellipse where \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). In this case, \(a^2 = 25\) and \(b^2 = 25\), meaning \(a = b = 5\).
4Step 4: Identify the Type of Conic Section
The rewritten equation is an ellipse. A circle is a special case of an ellipse where the length of the semi-major axis \(a\) is equal to the length of the semi-minor axis \(b\), so the equation reduces to \(\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\), which is equivalent to \(x^2 + y^2 = a^2\).
Key Concepts
Equation of a CircleEllipseStandard Form
Equation of a Circle
The equation of a circle is a mathematical representation that describes the set of all points equidistant from a central point, known as the center. A common form for this equation is shown as \(x^2 + y^2 = r^2\), where \(r\) is the radius, and the circle is centered at the origin (0,0).
This tells us that for a circle centered at the origin, any point (x,y) on the circle satisfies the equation. The equation ensures the distance from the center to any point on the circumference remains constant.
To illustrate, the exercise gives the equation \(x^2 + y^2 = 25\), meaning the circle has a radius of 5, because \(5^2 = 25\). This equation captures all the points 5 units away from the center at the origin.
This tells us that for a circle centered at the origin, any point (x,y) on the circle satisfies the equation. The equation ensures the distance from the center to any point on the circumference remains constant.
To illustrate, the exercise gives the equation \(x^2 + y^2 = 25\), meaning the circle has a radius of 5, because \(5^2 = 25\). This equation captures all the points 5 units away from the center at the origin.
Ellipse
An ellipse is another important conic section characterized by its stretched circular shape. Unlike a circle, an ellipse has two axes of symmetry: the major axis and the minor axis.
Its standard form equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. When \(a\) equals \(b\), the ellipse becomes a circle.
Its standard form equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. When \(a\) equals \(b\), the ellipse becomes a circle.
- The vertices of an ellipse give it its elongated shape.
- The foci, located along the major axis, influence the oval shape.
- If you move the equation's right side to be 1, as done in the exercise, you see how the equation \(\frac{x^2}{25} + \frac{y^2}{25} = 1\) is in the form of an ellipse. Here, \(a = b = 5\), forming a perfect circle.
Standard Form
The term 'standard form' refers to the primary equation format used to represent different conic sections, such as circles and ellipses.
For circles, the standard form is \(x^2 + y^2 = r^2\). This simplifies scenarios where the circle is centered at the origin. To transition to the general ellipse form, divide each term by \(r^2\), resulting in \(\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1\), aligning closely with the standard form for ellipses.
For circles, the standard form is \(x^2 + y^2 = r^2\). This simplifies scenarios where the circle is centered at the origin. To transition to the general ellipse form, divide each term by \(r^2\), resulting in \(\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1\), aligning closely with the standard form for ellipses.
- In any standard form equation, mathematical transformations or manipulations (e.g., completing the square) help identify and compare the shapes of conic sections.
- Standard form equations are fundamental because they provide a readily accessible framework to convert and examine mathematical expressions further—whether for solving geometry problems or analyzing more complex scenarios.
Other exercises in this chapter
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