Equations of circles questions with answers

Questions

1. Question 1Read the centre and radius

   Find the coordinates of the centre of the circle $(x + 2)^2 + (y - 1)^2 = 13$.

   Show the worked solution

   1. Step 1: (x − a)² + (y − b)² = r²: centre (a, b), radius r$a = -2$$b = 1$
2. Question 2Read the centre and radius

   Find the radius of the circle $(x + 3)^2 + (y + 1)^2 = 4$.

   Show the worked solution

   1. Step 1: (x − a)² + (y − b)² = r²: centre (a, b), radius r$r^2 = 4$$r = 2$
3. Question 3From the general form

   Find the coordinates of the centre of the circle $x^2 + y^2 + 8x + 6y - 11 = 0$.

   Show the worked solution

   1. Step 1: Group the x terms and the y terms, then complete the square in each$x^2 + 8x + y^2 + 6y = 11$$(x + 4)^2 - 16 + (y + 3)^2 - 9 = 11$$(x + 4)^2 + (y + 3)^2 = 36$
   2. Step 2: Read off the centre and radius$\text{Centre: } x = -4$$\text{Centre: } y = -3$
4. Question 4From the general form

   Find the coordinates of the centre of the circle $x^2 + y^2 + 2x - 8y + 8 = 0$.

   Show the worked solution

   1. Step 1: Group the x terms and the y terms, then complete the square in each$x^2 + 2x + y^2 - 8y = -8$$(x + 1)^2 - 1 + (y - 4)^2 - 16 = -8$$(x + 1)^2 + (y - 4)^2 = 9$
   2. Step 2: Read off the centre and radius$\text{Centre: } x = -1$$\text{Centre: } y = 4$
5. Question 5Find the equation

   A circle has centre $(2, 1)$ and passes through the point $(-1, 2)$. Find the equation of the circle.

   Show the worked solution

   1. Step 1: The centre is given$\text{Centre } (2, 1)$
   2. Step 2: r² = (distance from the centre to (-1, 2))²$r^2 = (-1 - 2)^2 + (2 - 1)^2$$r^2 = 10$
   3. Step 3: (x − a)² + (y − b)² = r²$(x - 2)^2 + (y - 1)^2 = 10$
6. Question 6Find the equation

   A circle has centre $(0, -4)$ and passes through the point $(5, 5)$. Find the equation of the circle.

   Show the worked solution

   1. Step 1: The centre is given$\text{Centre } (0, -4)$
   2. Step 2: r² = (distance from the centre to (5, 5))²$r^2 = (5 - 0)^2 + (5 - (-4))^2$$r^2 = 106$
   3. Step 3: (x − a)² + (y − b)² = r²$x^2 + (y + 4)^2 = 106$