Question

1. The beginning steps for determining the center and radius of a circle using the completing the square method are shown below:

Answer 1

(1) Answer : $(x^2 - 6x + 9) + (y^2- 4y + 4) = 3 + (9 + 4)$

Step 1: $x^2 - 6x + y^2 - 4y = 3$

Step 2: $(x^2- 6x) + (y^2 - 4y) = 3$

In completing the square method we take coefficient of x and divide by 2 and the square it . Then add it on both sides

The coefficient of x is -6. $\frac{-6}{2}$ = (-3)^2 = 9

The coefficient of y is -4. $\frac{-4}{2}$ = (-2)^2 = 4

Step : $(x^2- 6x + 9) + (y^2 - 4y + 4) = 3 +9 + 4$

(2) $x^2 + y^2 + 6x - 6y + 2 = 0$

To find center and radius we write the equation in the form of

$(x-h)^2 + (y-k)^2 = r^2$ using completing the square form

Where (h,k) is the center and 'r' is the radius

$x^2 + y^2 + 6x - 6y + 2 = 0$

$(x^2 + 6x) + (y^2 - 6y) + 2 = 0$

In completing the square method we take coefficient of x and divide by 2 and the square it . Then add it on both sides

$(x^2 + 6x + 9) + (y^2 - 6y + 9) = -2 + 9 + 9$

$(x + 3)^2 + (x - 3)^2 = 16$

Here h= -3 and k=3 and $r^2 = 16$ so r= 4

Center is (-3,3) and radius = 4

(c) Step 1: $x^2 + 8x + y^2 - 6y = 11$

Step 2: $(x^2 + 8x) + (y^2 - 6y) = 11$

Step 3: $(x^2 + 8x + 16) + (y^2 - 6y + 9) = 11 + (16 + 9)$

Step 4: $(x^2 + 8x + 16) + (y^2 - 6y + 9) = 36$

We factor out each quadratic

(x^2 + 8x + 16) = (x+4)(x+4) = $(x+4)^2$

((y^2 - 6y + 9)) = (x-3)(x-3) = $(x-3)^2$

Step 5 :$(x + 4)^2 + (y - 3)^2 = 6^2$