Problem 71 - College Algebra

Question

Which one of the following is true? a. The equation of the circle whose center is at the origin with radius 16 is \(x^{2}+y^{2}=16\) b. The graph of \((x-3)^{2}+(y+5)^{2}=36\) is a circle with radius 6 centered at \((-3,5)\) c. The graph of \((x-4)+(y+6)=25\) is a circle with radius 5 centered at \((4,-6)\) d. None of the above is true.

Step-by-Step Solution

Answer

The correct answer is d. None of the above is true.

Step 1: Analyze option a

Option a states that the equation of the circle is \(x^{2}+y^{2}=16\). This is clearly a circle with center at the origin, but its radius should be the square root of 16 which is 4 not 16. So, option a is incorrect.

Step 2: Analyze option b

Option b says the graph of \((x-3)^{2}+(y+5)^{2}=36\) is a circle with radius 6 centered at \((-3,5)\). The equation fits the standard form of a circle's equation, and the radius \(r\) is indeed equal to the square root of 36, which is 6. However, the center of the circle should be \((3,-5)\), not \((-3,5)\) Hence, option b is incorrect.

Step 3: Analyze option c

Option c suggests the graph of \((x-4)+(y+6)=25\) is a circle with radius 5 centered at \((4,-6)\). The equation does not fit the standard form of a circle's equation, which is \((x-h)^{2}+(y-k)^{2}=r^{2}\). Hence, option c is incorrect.

Step 4: Check the last option

As options a, b, and c are all incorrect, it means that the last option which is 'None of the above is true' should be correct.

Key Concepts

Standard Form of a Circle

Understanding the standard form of a circle equation is fundamental to dealing with circles in algebra. The standard form equation of a circle is expressed as \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) represents the center of the circle and \( r \) is the radius. The terms \( (x-h) \) and \( (y-k) \) show a horizontal and vertical shift from the origin, respectively.

When the equation is not in this form, as we observed with one of the options in the exercise, it cannot represent a circle. This is an important concept to remember, as it helps to quickly identify whether an equation is that of a circle or not. Additionally, bringing other equations into this standard form can help to simplify the process of understanding their graphs.

Radius of a Circle

The radius of a circle is a straight line from the center to any point on the circle's circumference. It is half the diameter and is a key component in the geometry of a circle. In the context of the circle equation \( (x-h)^2 + (y-k)^2 = r^2 \), the value of \( r^2 \) often confuses students—they sometimes mistakenly take the given \( r^2 \) as the radius. However, the radius is the square root of this value.

For example, an equation like \( x^2 + y^2 = 16 \) might tempt one to say the radius is 16, when it is actually 4, the square root of 16. Recognizing this error is crucial for accurately determining the properties of a circle from its equation.

Center of a Circle

The center of a circle is its midpoint and is denoted by the coordinates \( (h, k) \) in the standard form equation. To find the center when given a circle's equation, look for the values of \( h \) and \( k \) in the terms \( (x-h)^2 \) and \( (y-k)^2 \) respectively. A common mistake is to misidentify the signs; \( (x-h)^2 \) indicates the center \( h \) has the opposite sign of what appears in the equation; the same applies to \( k \) from \( (y-k)^2 \).

For instance, the equation \( (x-3)^2 + (y+5)^2 = 36 \) indicates a center at \( (3, -5) \) rather than \( (-3, 5) \) as the signs are inverted inside the brackets. This detail can make all the difference in correctly graphing the circle.