/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 We know that \(x^{2}+y^{2}=25\) ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 50

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.

Short Answer

Expert verified
The equation \(x^2 + y^2 = 25\) becomes \(\frac{x^2}{25} + \frac{y^2}{25} = 1\), resembling an ellipse. It is a circle when \(a = b\).

Step by step solution

01

Understand the Equation

The given equation \(x^2 + y^2 = 25\) represents a circle centered at the origin with a radius of 5. The equation is in the general form of a circle \((x - h)^2 + (y - k)^2 = r^2\), where \(h = 0\), \(k = 0\), and \(r = 5\).
02

Rewrite the Equation

To rewrite the equation such that the right side equals 1, divide the entire equation by 25. This simplifies to \(\frac{x^2}{25} + \frac{y^2}{25} = 1\).
03

Identify the Conic Section

The rewritten equation resembles the general form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a = 5\) and \(b = 5\). An ellipse is defined by this equation, which shows the relationship between \(x\) and \(y\) for ellipses.
04

Identify Ellipse Conditions for a Circle

For \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) to represent a circle, the denominators of \(x^2\) and \(y^2\) must be equal, meaning \(a = b\). This shows that when both semi-axis lengths (\(a\) and \(b\)) are equal, the ellipse is a circle. In this specific case, with \(a = 5\) and \(b = 5\), the conic is a circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Equation
The circle equation is a crucial concept in geometry. It describes all the points that form a circle in a coordinate system. For any circle centered at the origin, the equation takes the form \[ x^2 + y^2 = r^2 \], where
  • \( x \) and \( y \) are the coordinates of any point on the circle, and
  • \( r \) is the radius of the circle.
This equation tells us that the distance from the center of the circle to any point on the circumference is always \( r \). When you see an equation like \( x^2 + y^2 = 25 \), you can instantly recognize it as a circle with a radius of 5, since \( r^2 = 25 \).
Ellipse Conditions
Ellipses are a type of conic section characterized by the equation:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]This equation shares a resemblance to the circle's equation but introduces two different parameters, \( a \) and \( b \), which represent the lengths of the semi-major and semi-minor axes, respectively.
An important condition under which an ellipse becomes a circle is when the lengths of its semi-major and semi-minor axes are equal, i.e., \( a = b \). When this condition is met, the ellipse equation transforms into the equation of a circle, reflecting the symmetry in all directions from the center.
Standard Form of Conic Sections
Conic sections are curves obtained from intersecting a plane with a cone. The standard form of the equations for the different conic sections varies. For circles and ellipses, the standard forms are:
  • Circle: \[ (x - h)^2 + (y - k)^2 = r^2 \],where \((h, k)\) is the circle's center.
  • Ellipse: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \],where \((h, k)\) is the center, and \(a\) and \(b\) are the axis lengths.
The transition from a circle to an ellipse happens purely when \(a eq b\) while keeping the equation balanced as shown.
Radius of a Circle
The radius of a circle is the distance from its center to any point on its circumference. It provides vital information about the size of the circle. In the equation \[ x^2 + y^2 = r^2 \], \( r \) is clearly the radius.
In our example where the equation is \( x^2 + y^2 = 25 \), you find the radius by calculating the square root of 25, which yields \( r = 5 \).
Understanding radius helps you easily define the circle's boundary and comprehend its symmetry and area.

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Most popular questions from this chapter

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$x^{2}+y^{2}=1$$

Sketch the graph \(\frac{(x+5)^{2}}{16}-\frac{(y+2)^{2}}{25}=1\)

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$x^{2}+(y+5)^{2}=5$$

Rationalize each denominator and simplify, if possible. See Section 10.5 $$ \frac{10}{\sqrt{5}} $$

The graph of each equation is a parabola. Find the vertex of the parabola and sketch its graph. See Examples I through 4. $$ x=(y-4)^{2}-1 $$
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