Question

How do you graph the circle with center at \[\left( {4,0} \right)\] and radius 3 and label the center and at least four points on the circle, then write the equation of the circle?

Answer 1

Hint: In this question we are asked to plot the circle with centre and radius given, We first Plot the center of the circle at \[\left( {h,k} \right)\] and then count out from the center \[r\] units in the four directions (up, down, left, and right). Then, connect those four points with a nice, round circle, and here \[h = 4\], \[k = 0\], and \[r = 3\] substitute the center and radius in the standard form of equation of circle which is given by \[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\].

Complete step by step solution:
A circle is the set of all points the same distance from a given point, the center of the circle. A radius, \[r\], is the distance from that center point to the circle itself.
Now plotting the circle on the graph,
Given center \[\left( {4,0} \right)\] and radius 3,
Place the center of the circle at \[\left( {4,0} \right)\], which is shown in the figure below,

Plot the radius points on the coordinate plane, which is given as 3,

Now count 3 units up, down, left, and right from the center at \[\left( {4,0} \right)\]to plot the four points on the circle
Count 3 units right, i.e., add 3 units to the \[x\]-coordinate of the center we get,
\[ \Rightarrow \left( {4 + 3,0 + 0} \right) = \left( {7,0} \right)\],
Now count 3 units left i.e., subtract 3 units from the \[x\]-coordinate of the center we get,
\[ \Rightarrow \left( {4 - 3,0 + 0} \right) = \left( {1,0} \right)\],
Now count 3 units up, i.e., add 3 units to the \[y\]-coordinate of the center we get,
\[ \Rightarrow \left( {4 + 0,0 + 3} \right) = \left( {4,3} \right)\],
Now count 3 units down, subtract 3 units to the \[y\]-coordinate of the center we get,\[ \Rightarrow \left( {4 + 0,0 - 3} \right) = \left( {4, - 3} \right)\],
 This means that we should have points at, \[\left( {7,0} \right)\], \[\left( {1,0} \right)\], \[\left( {4,3} \right)\] and \[\left( {4, - 3} \right)\].
Connect the dots to the graph of the circle with a round, smooth curve.

Now we know that the equation of circle with center \[\left( {h,k} \right)\]and radius \[r\]is given by \[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\], so here center \[\left( {4,0} \right)\] and radius 3, substituting the values in the equation we get,
\[ \Rightarrow {\left( {x - 4} \right)^2} + {\left( {y - 0} \right)^2} = {3^2}\],
Now simplifying we get,
\[ \Rightarrow {\left( {x - 4} \right)^2} + {y^2} = 9\].
So, the equation of the circle is \[{\left( {x - 4} \right)^2} + {y^2} = 9\].

\[\therefore \] The required graph for the circle with center \[\left( {4,0} \right)\]and radius 3 will be shown as,

and the equation of circle with center \[\left( {4,0} \right)\]and radius 3 will be equal to\[{\left( {x - 4} \right)^2} + {y^2} = 9\].

Note:
Graphing circles requires two things: the coordinates of the center point, and the radius of a circle. While plotting the circle on the graph we have to remember to switch the sign of the \[h\] and \[k\]from inside the parentheses in the equation. This is necessary because the \[h\] and \[k\] are inside the grouping symbols, which means that the shift happens opposite from what we would think.