Question

How do you graph ${x^2} + {y^2} - 6x + 8y + 9 = 0$ ?

Answer 1

Hint: In this question, we will try to rearrange the terms in the equation and try to get it in the form of a standard circle equation. Then, from that equation, we will take the points of the center and the radius and using them, we will plot the graph.

Formula used: Standard equation of a circle: ${(x - a)^2} + {(y - b)^2} = {r^2}$
Where $(a,b)$ is the center of the circle and $r$ is the radius of the circle.

Complete step-by-step solution:
We have the equation given as:
${x^2} + {y^2} - 6x + 8y + 9 = 0$
Now, on grouping the similar terms and shifting the constant value on the right-hand side, we get:
$\Rightarrow$$({x^2} - 6x) + ({y^2} + 8y) = - 9$
Now, to get the equation in the form of a standard circle equation, we will complete the squares by adding numbers on both sides of the equation.
Now the third term should be such that it is the square of the coefficient of $x$ divided by $2$ . Therefore, we will add ${\left( {\dfrac{6}{2}} \right)^2}$ which is ${3^2}$ and ${\left( {\dfrac{8}{2}} \right)^2}$ which is ${4^2}$ on both the sides of the equation.
On completing the squares, we get:
$\Rightarrow$$({x^2} - 6x + {3^2}) + ({y^2} + 8y + {4^2}) = - 9 + {3^2} + {4^2}$
We know that ${(a + b)^2} = {a^2} + 2ab + {b^2}$ . Therefore, on writing the square in the simplified manner, we get:
$\Rightarrow$${(x - 3)^2} + {(y + 4)^2} = - 9 + {3^2} + {4^2}$
Now we know that ${3^2} = 9$ therefore, on simplifying the right-hand side we get:
$\Rightarrow$${(x - 3)^2} + {(y + 4)^2} = {4^2}$
Now the standard equation of the circle is: ${(x - a)^2} + {(y - b)^2} = {r^2}$
Therefore, the above expression can be also be written as:
$\Rightarrow$${(x - 3)^2} + {(y - ( - 4))^2} = {4^2}$
On comparing the equation with the equation of a standard circle, we get:
The center of the circle is $(3, - 4)$ and the radius of the circle is $4$ .
The circle can be plotted on the graph as:

Where point $A$ is the center of the circle and the distance from point $A$ to $B$ is the radius of the circle.

Note: It is to be remembered that adding and subtracting values on both sides of the $ = $ sign, the value of the equation does not change.
The equation of a circle with its center at the origin is generally used, the equation is ${x^2} + {y^2} = {r^2}$ , where $r$ is radius of the circle and $\left( {0,0} \right)$ is the center of the circle.