Tutorials with detailed solutions to examples and matched exercises on finding equation of a circle, radius and center. Detailed explanations are also provided.
A circle is the set of points equidistant from a point \( C(h,k) \) called the center. The fixed distance \( r \) from the center to any point on the circle is called the radius.
The standard equation of a circle with center at \( C(h,k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Find the equation of a circle whose center is at \((2, -4)\) and radius \(5\).
Substitute \((h, k)\) by \((2, -4)\) and \(r\) by \(5\) in the standard equation to obtain:
\[ (x - 2)^2 + (y - (-4))^2 = 5^2 \]
Simplify:
\[ (x - 2)^2 + (y + 4)^2 = 25 \]
Set \(h\), \(k\), and \(r\) parameters into this applet and plot the circle. Verify graphically that the equation is that of the circle with the given center and radius.
Matched Exercise 1
Find the equation of a circle whose center is at \((2, -4)\) and radius \(3\).
Find the equation of a circle that has a diameter with the endpoints given by the points \(A(-1, 2)\) and \(B(3, 2)\).
The center \(C\) of the circle is the midpoint of the line segment making the diameter \(AB\). We first use the midpoint formula to find the coordinates of \(C\):
\[ C\left( \frac{{(-1 + 3)}}{2}, \frac{{(2 + 2)}}{2} \right) = C(1,2) \]
The radius \(r\) is half the distance between \(A\) and \(B\). Hence:
\[ r = \frac{1}{2} \sqrt{ [3 - (-1)]^2 + [2 - 2]^2 } \]
\[ = \frac{1}{2} \sqrt{4^2 + 0^2} = 2 \]
The coordinates of \(C\) and the radius \(r\) are used in the standard equation of the circle to obtain the equation:
\[ (x - 1)^2 + (y - 2)^2 = 2^2 \]
Simplify:
\[ (x - 1)^2 + (y - 2)^2 = 4 \]
Set the \(h\), \(k\) and \(r\) parameters into this applet and plot the circle. Verify graphically that the equation is that of a circle with the diameter as given above.
Matched Exercise 2
Find the equation of a circle that has a diameter with the endpoints given by \(A(0, -2)\) and \(B(0, 2)\).
Find the center and radius of the circle with equation:
\[ x^2 - 4x + y^2 - 6y + 9 = 0 \]
In order to find the center and the radius of the circle, we first rewrite the given equation into the standard form. Put all terms with \(x\) and \(x^2\) together and all terms with \(y\) and \(y^2\) together using parentheses:
\[ (x^2 - 4x) + (y^2 - 6y) + 9 = 0 \]
We now complete the square within the parentheses:
\[ (x^2 - 4x + \color{red}{4}) - \color{red}{4} + (y^2 - 6y + \color{red}{9}) - \color{red}{9} + 9 = 0 \]
Which may be written as:
\[ (x - 2)^2 + (y - 3)^2 - 4 - 9 + 9 = 0 \]
Simplify and write in standard form:
\[ (x - 2)^2 + (y - 3)^2 = 4 \]
\[ (x - 2)^2 + (y - 3)^2 = 2^2 \]
We now compare this equation and the standard equation to obtain:
\[ \text{Center at } C(h, k) = C(2, 3) \quad \text{and} \quad \text{radius } r = 2 \]
Matched Exercise 3
Find the center and radius of the circle with equation:
\[ x^2 - 2x + y^2 - 8y + 1 = 0 \]
Is the point \(P(3, 4)\) inside, outside or on the circle with equation:
\[ (x + 2)^2 + (y - 3)^2 = 9 \]
We first find the distance from the center of the circle to point \(P\). Using the given equation the center \(C\) is at \((-2, 3)\) and the radius \(r = \sqrt{9} = 3\).
Distance from \(C\) to \(P\) is equal to:
\[ \sqrt{[3 - (-2)]^2 + [4 - 3]^2} = \sqrt{5^2 + 1^2} = \sqrt{26} \approx 5.1 \]
Since the distance from \(C\) to \(P\) is \(\sqrt{26} \approx 5.1\) which is greater than the radius \(r = 3\), point \(P\) is outside the circle.
You can check your answer graphically using this applet.
Matched Exercise 4
Is the point \(P(-1, -3)\) inside, outside or on the circle with equation:
\[ (x - 1)^2 + (y + 3)^2 = 4 \]
Find the equation of the circle such that the three points \(A(0, 4)\), \(B(3, 5)\) and \(D(7, 3)\) are on the circle.
The distance from the center \(C(h, k)\) of the circle to each of the points \(A\), \(B\) and \(D\) is constant and equal to the radius \(r\) of the circle. Write three equations stating that these distances are equal to the radius \(r\):
\[ d(A,C) = \sqrt{(h - 0)^2 + (k - 4)^2} = r \]
\[ d(B,C) = \sqrt{(h - 3)^2 + (k - 5)^2} = r \]
\[ d(D,C) = \sqrt{(h - 7)^2 + (k - 3)^2} = r \]
Write that \(d(A,C) = d(B,C)\) and \(d(A,C) = d(D,C)\):
\[ \sqrt{(h - 0)^2 + (k - 4)^2} = \sqrt{(h - 3)^2 + (k - 5)^2} \]
\[ \sqrt{(h - 0)^2 + (k - 4)^2} = \sqrt{(h - 7)^2 + (k - 3)^2} \]
Square each side of each equation:
\[ (h - 0)^2 + (k - 4)^2 = (h - 3)^2 + (k - 5)^2 \]
\[ (h - 0)^2 + (k - 4)^2 = (h - 7)^2 + (k - 3)^2 \]
Expand the squares in the above equations and simplify:
\[ -8k + 16 = -6h + 9 - 10k + 25 \]
\[ -8k + 16 = -14h + 49 - 6k + 9 \]
Write the above system of equations in standard form:
\[ \begin{cases} 2k + 6h = 18 \\ -2k + 14h = 42 \end{cases} \]
Use the method of addition to solve the system:
\[ 20h = 60 \quad \Rightarrow \quad h = 3 \]
Substitute \(h\) by its value \(3\) in one of the equations to obtain \(k\):
\[ k = 0 \]
We now use one of the distance formulas above to find the radius \(r\):
\[ r = \sqrt{(3 - 0)^2 + (0 - 4)^2} = 5 \]
The equation of the circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
\[ (x - 3)^2 + y^2 = 25 \]
Shown below is the graph of the circle with the three points:
Matched Exercise 5
Find the equation of the circle such that the three points \(A(-5, 0)\), \(B(1, 0)\) and \(D(-2, -3)\) are on the circle.
Find the equation of the circle that is tangent to the line whose equation is given by \(x + y = 2\) and has its center at \((3, 5)\).
The first step is to determine the point of tangency of the circle and the line \(x + y = 2\). Use the property of circles that a line through the center \(C\) of a circle and the point of tangency \(T\) (let us call this line \(CT\)) and the line \(x + y = 2\) (let us call this line \(LT\)) tangent to the circle are perpendicular (see graph below).
The slopes \(m_1\) and \(m_2\) of two perpendicular lines are related by the formula: \(m_1 \times m_2 = -1\). Find the slope \(m_1\) of line \(LT\):
\[ x + y = 2 \quad \Rightarrow \quad y = -x + 2 \quad \Rightarrow \quad m_1 = -1 \]
We now use the formula to find the slope \(m_2\) of line \(CT\):
\[ m_2 = -\frac{1}{m_1} = 1 \]
The equation of the line \(CT\) which passes through the center \(C(3, 5)\) is given by:
\[ y - 5 = m_2 (x - 3) \quad \Rightarrow \quad y = x + 2 \]
The point of tangency is the intersection of lines \(CT\) and \(LT\) and is found by solving the system of equations:
\[ \begin{cases} x + y = 2 \\ y = x + 2 \end{cases} \]
The point of tangency is at \((0, 2)\).
The distance between the center of the circle and the point of tangency is equal to the radius \(r\) of the circle and is given by:
\[ r = \sqrt{(3 - 0)^2 + (5 - 2)^2} = 3\sqrt{2} \]
Let \(h\) and \(k\) be the \(x\) and \(y\) coordinates of the center of the circle and \(r\) its radius, the equation of the circle in standard form is:
\[ (x - 3)^2 + (y - 5)^2 = (3\sqrt{2})^2 \]
\[ (x - 3)^2 + (y - 5)^2 = 18 \]
Shown below is the graph of the circle and the line \(x + y = 2\) tangent to it:
Matched Exercise 6
Find the equation of the circle that is tangent to the line whose equation is given by \(x + 2y = 2\) and has its center at \((0,5)\).