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Chapter 2: Problem 80

Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.

Short Answer

Expert verified
No, \((x-3)^{2}+(y-5)^{2}=0\) does not represent the equation of a circle with a nonzero radius. The equation represents a point located at (3,5) in the coordinate plane.

Step by step solution

01

Identify the form of the equation

The given equation \((x-3)^{2}+(y-5)^{2}=0\) is of the form \((x-h)^{2}+(y-k)^{2}=r^{2}\). Here, \(h=3\), \(k=5\), and \(r^{2}=0\). This is the standard form of the equation of a circle where \((h, k)\) is the center and \(r\) is the radius.
02

Determine the values of h, k, and r

From the standard form, \(h=3\), \(k=5\) and \(r^{2}=0\). Taking the square root of \(r^{2}\), we find that \(r=0\).
03

Describe the graph

Since the radius of the circle is 0, the circle degenerates to a single point. Therefore, the graph of the equation will be a single point at the location \((h,k)\) or in this case, \((3,5)\).

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Most popular questions from this chapter

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+2)^{2}+(y+2)^{2}=4 $$

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}+6 x+2 y+6=0 $$

determine whether each statement makes sense or does not make sense, and explain your reasoning. A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point \((3,-4)\)

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} (x-3)^{2}+(y+1)^{2} &=9 \\ y &=x-1 \end{aligned} $$

Explain how to identify the domain and range of a function from its graph.
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