Answer

Verified

416.7k+ views

**Hint:**First we will evaluate the centre of the circle using the standard equation of circle ${(x - a)^2} + {(y - b)^2} = {r^2}$, where $(a,b)$ is the centre of the circle and $r$ is the radius of the circle. Use the midpoint formula to evaluate the centre of the circle and hence the radius of the circle.

**Complete step-by-step solution:**

The general equation of circle is given by ${(x - a)^2} + {(y - b)^2} = {r^2}$

Where, $(a,b)$ is the centre of the circle and $r$ is the radius of the circle.

Now we will convert the equation ${x^2} + 6x + {y^2} - 2y + 6 = 0$ into the form ${(x - a)^2} + {(y - b)^2} = {r^2}$.

$

\Rightarrow {x^2} + 6x + {y^2} - 2y + 6 = 0 \\

\Rightarrow ({x^2} + 6x + 9) + {y^2} - 2y + 6 = 9 \\

\Rightarrow {(x + 3)^2} + {y^2} - 2y + 6 = 9 \\

$

Now we will subtract $6$ from both the sides.

$

\Rightarrow {(x + 3)^2} + {y^2} - 2y + 6 = 9 \\

\Rightarrow {(x + 3)^2} + {y^2} - 2y = 3 \\

$

Now we will complete the square of the $y$ terms.

$

\Rightarrow {(x + 3)^2} + {y^2} - 2y = 3 \\

\Rightarrow {(x + 3)^2} + ({y^2} - 2y + 1) = 3 + 1 \\

\Rightarrow {(x + 3)^2} + {(y - 1)^2} = 4 \\

$

First, we will evaluate the centre of the circle using the standard equation of circle ${(x - a)^2} + {(y - b)^2} = {r^2}$, where $(a,b)$ is the centre of the circle and $r$ is the radius of the circle

Now we will compare and then evaluate the coordinates of the centre ${(x + 3)^2} + {(y - 1)^2} = 4$

Hence, the centre of the circle is $( - 3,1)$.

Now we substitute all these values in the general equation of circle which is given by ${(x - a)^2} + {(y - b)^2} = {r^2}$

$

\Rightarrow {(x + 3)^2} + {(y - 1)^2} = {(2)^2} \\

\Rightarrow {(x + 3)^2} + {(y - 1)^2} = 4 \\

$

Hence, the radius of the circle is $4\,units$.

**Note:**While solving for the centre of the circle, substitute the values in the midpoint formula along with their signs. While comparing values of terms with the general equation, compare along with their respective signs. Substitute values carefully in the formulas for evaluating values of terms. While evaluating the value of the coordinates pay attention to the signs of the terms and also first compare the terms.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers