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Question: How do you write the standard form of the equation of the circle with center \(\left( {2,3} \right)\...

How do you write the standard form of the equation of the circle with center (2,3)\left( {2,3} \right) and radius 4?

Explanation

Solution

Here, we are required to write the standard form of the equation of the circle whose center is (2,3)\left( {2,3} \right) and radius is 4 units. Thus, we will write the standard form of the equation of the circle and simply substitute the given coordinates of the center and the given radius to find the required stand equation.

Formula Used:
The standard equation of a circle is: (x−h)2+(y−k)2=r2{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}
Where, (h,k)\left( {h,k} \right) are the coordinates of the center of the circle and rrrepresents the radius of the circle.

Complete step by step solution:
As we know, the standard equation of a circle is:
(x−h)2+(y−k)2=r2{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}
Where, (h,k)\left( {h,k} \right) are the coordinates of the center of the circle.
And, rr represents the radius of the circle
Now, according to the question, we are given that the center of the circle is (2,3)\left( {2,3} \right)
Therefore, substituting (h,k)=(2,3)\left( {h,k} \right) = \left( {2,3} \right), we get,
(x−2)2+(y−3)2=r2{\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} = {r^2}
Also, it is given that the radius of the circle is 4 units.
Thus, further substituting r=4r = 4, we get,
(x−2)2+(y−3)2=42{\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} = {4^2}
⇒(x−2)2+(y−3)2=16\Rightarrow {\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} = 16
Clearly, this is the standard form of the equation of a circle.

Thus, the standard form of the equation of the circle with center (2,3)\left( {2,3} \right) and radius 4 is (x−2)2+(y−3)2=16{\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} = 16
Hence, this is the required answer.

Note:
A circle is a round shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape. We know that the general equation for a circle is (x−h)2+(y−k)2=r2{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}, where (h,k)\left( {h,k} \right) is the centre and rr is the radius. Circle has various properties such as it is the shape with the largest area for a given length of perimeter. The circle is a highly symmetric shape. Also, every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle.