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Chapter 40: Problem 859

Discuss the graph of the equation \(\mathrm{x}^{2}+\mathrm{y}^{2}=25\).

Short Answer

Expert verified
The given equation \(x^2 + y^2 = 25\) represents a circle centered at the origin (0, 0) with a radius of 5 units. Notable points on the graph include intersections with the x-axis at (5, 0) and (-5, 0), intersections with the y-axis at (0, 5) and (0, -5), and intersections with the lines y = x and y = -x at \(\pm\sqrt{12.5}, \pm\sqrt{12.5}\). The graph exhibits symmetry with respect to the x-axis, y-axis, and the origin.

Step by step solution

01

Determine the type of graph

The given equation, \(x^2 + y^2 = 25\), is in the form of a circle with the equation \((x-h)^2 + (y-k)^2 = r^2\) where (h, k) is the center of the circle and r is the radius. In our case, h = 0 and k = 0, as there is no shifting of the x and y variables. This means that our equation represents a circle centered at the origin (0, 0).
02

Find the radius of the circle

The given equation can also be written as \((x-0)^2 + (y-0)^2 = 25\), which means r^2 = 25. To find the radius, take the square root of 25: \[ r = \sqrt{25} = 5 \] So, the radius of the circle is 5 units.
03

Note any identifiable points on the graph

With a circle centered at the origin (0, 0) and a radius of 5, we can identify some specific points on the graph: 1. (5, 0) - the circle intersects the x-axis on the positive side 2. (-5, 0) - the circle intersects the x-axis on the negative side 3. (0, 5) - the circle intersects the y-axis on the positive side 4. (0, -5) - the circle intersects the y-axis on the negative side In addition, we can identify when x = y: 5. \(\pm\sqrt{12.5}, \pm\sqrt{12.5}\) - the circle intersects the lines y = x and y = -x
04

Discuss the symmetry of the graph

Since the circle is centered at the origin (0, 0) with radius 5, the graph is symmetrical with respect to: 1. The x-axis: The points (x, y) and (x, -y) on the circle have the same x-coordinates. 2. The y-axis: The points (x, y) and (-x, y) on the circle have the same y-coordinates. 3. The origin: The points (x, y) and (-x, -y) on the circle are symmetrical with respect to the origin. In conclusion, the given equation \(x^2 + y^2 = 25\) represents a circle centered at the origin (0, 0) with a radius of 5 units. The notable points on the graph include the intersections with the x-axis, y-axis, and the lines y = x and y = -x. The graph exhibits symmetry with respect to the x-axis, y-axis, and the origin.

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Key Concepts

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

Graphing Circles
Graphing a circle can be a fun and easy task once you understand the basic formula for a circle, \((x-h)^2 + (y-k)^2 = r^2\). Here,
  • \((h, k)\) is the center of the circle
  • \(r\) is the radius of the circle
In our exercise, the equation \(x^2 + y^2 = 25\) reveals that the circle is centered at the origin (0, 0). This makes the math straightforward and the graph easy to plot. With the given radius of 5, you can easily sketch the circle. Simply draw a round shape with all points at an equal distance of 5 units from the origin.
Once you have marked these points, connect them smoothly to create the circle.
Symmetry in Graphs
Symmetry plays a vital role in understanding and simplifying graphing. For the equation \(x^2 + y^2 = 25\), symmetry is present in three distinct ways:
  • X-axis Symmetry: The graph looks the same on the top and bottom of the x-axis. Coordinate pairs \((x, y)\) and \((x, -y)\) are mirror images.
  • Y-axis Symmetry: You鈥檒l notice that the graph mirrors itself on the left and right sides of the y-axis, meaning each \((x, y)\) point has a \((-x, y)\) counterpart.
  • Origin Symmetry: The circle is also symmetrical about the origin; each point \((x, y)\) has a \((-x, -y)\) reflecting back through the center.
This symmetrical nature helps in better predicting where points will fall on your graph.
Radius of a Circle
The radius of a circle is crucial as it defines the size of the circle. It is the distance from the center to any point on the circle's perimeter. In the equation \(x^2 + y^2 = 25\), \(r^2 = 25\). Finding the radius means solving \(r = \sqrt{25}\), which gives us \(r = 5\).
The radius helps in graphing and maintaining the circle鈥檚 perfect round shape.
When graphing, if you know the center and radius, just draw all points that are 5 units away from \((0, 0)\). These points, once connected, complete the circle.
Center of a Circle
The center of a circle is the point from which all points on the circle are equidistant. In our equation \(x^2 + y^2 = 25\), no shifting is needed, meaning the circle is centered at the origin \((0, 0)\). This is the simplest scenario for a circle, removing any complexity of translation.
Having the circle at the origin makes calculations straightforward.
It is a foundational point for symmetry and for determining intersections with axes. Knowing the center not only helps in graphing but also in understanding deeper properties of the circle.

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Most popular questions from this chapter

Find the equation of the circle having radius 4 , tangent to the line \(1: x=-y\) at \((0,0)\).

Find the center and radius of the circle \(x^{2}-4 x+y^{2}+8 y-5=0\).

Find the intersection between the circles $$ x^{2}+y^{2}=4 \text { and } x^{2}+y^{2}-8 y+12=0. $$

Find the equation of the circle that goes through the points \((1,2)\) and \((3,4)\) and has radius \(\mathrm{a}=2\).

Given that two circles \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{D}_{1} \mathrm{x}+\mathrm{E}_{1} \mathrm{y}+\mathrm{F}_{1}=0\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{D}_{2} \mathrm{x}+\mathrm{E}_{2} \mathrm{y}+\mathrm{F}_{2}=0\) intersect at two points, show that the equation for the line determined by the points of intersection is \(\left(\mathrm{D}_{1}-\mathrm{D}_{2}\right) \mathrm{x}+\left(\mathrm{E}_{1}-\mathrm{E}_{2}\right) \mathrm{y}+\left(\mathrm{F}_{1}-\mathrm{F}_{2}\right)=0\).
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