Question

How do you find the center and radius of the circle \[{{x}^{2}}-2x+{{y}^{2}}-8y+1=0\]?

Answer 1

Hint: The general form of the equation of circle is \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]. We can see that coefficients of the square terms are one in the above equation. We can find the coordinates of the center of the circle and the radius using the values of the coefficient of the equation. The coordinates of the center of the circle are \[\left( -g,-f \right)\]. The radius of the circle is \[\sqrt{{{g}^{2}}+{{f}^{2}}-c}\].

Complete step by step solution:
We are given the equation of the circle \[{{x}^{2}}-2x+{{y}^{2}}-8y+1=0\]. Comparing the given equation with the general form of the equation of the circle. We get the values of g, f, and c as \[-1,-4\And 1\] respectively. We know that for a circle represented by the general form of the equation, the X and Y coordinates of the center are \[-g,-f\] respectively. Thus, for the given circle, by substituting the values of g and f, we get the coordinates of the center as \[\left( -(-1),-(-4) \right)\]. Simplifying this, we get center coordinates \[\left( 1,4 \right)\].
The radius of the circle is \[\sqrt{{{g}^{2}}+{{f}^{2}}-c}\], substituting the values of g, f, and c, we get radius as \[\sqrt{{{1}^{2}}+{{4}^{2}}-1}=4\].
We can also use the equation to graph the circle as,

Note: We can also use the center radius form of the equation of circle as follows,
\[{{x}^{2}}-2x+{{y}^{2}}-8y+1=0\]
Adding 16 to both sides of the equation, we get
\[\Rightarrow {{x}^{2}}-2x+{{y}^{2}}-8y+1+16=16\]
rearranging the terms of above equation, it can be written as
\[\Rightarrow \left( {{x}^{2}}-2x+1 \right)+\left( {{y}^{2}}-8y+16 \right)=16\]
We know that 16 is a square of 4. Using the algebraic expansions of \[{{\left( a+b \right)}^{2}}\And {{\left( a-b \right)}^{2}}\], the above equation can also be expressed as
\[\Rightarrow {{\left( x-1 \right)}^{2}}+{{\left( y-4 \right)}^{2}}={{4}^{2}}\]
Comparing with the centre radius form of circle, we get centre as \[\left( 1,4 \right)\] and radius as 4