Question

Fill in the boxes with numbers and operations to reflect the equation of the circle after the transformations

Answer 1

Given the equation of the circle:

`x^2+4x+y^2-12y=-24`

We will complete the squares for x and y to write the equation of the circle as:

`(x-h)^2+(y-k)^2=r^2`

Where (h,k) is the coordinates of the circle and r is the radius of the circle

so, the equation will be:

`\begin{gathered} (x^2+4x+4)+(y^2-12y+36)=-24+4+36 \\ \\ (x+2)^2+(y-6)^2=16 \\ \\ (x+2)^2+(y-6)^2=4^2 \end{gathered}`

So, the radius of the circle = 4

And the center of the circle = ( -2, 6 )

Now, we will make the transformation for the circle:

Shift 1 unit up, the center will be = ( -2, 7 )

shift 2 units right, the center will be = ( 0, 7 )

Reflection across the x-axis, the center will be = ( 0, -7)

So, the equation of the circle after transformation will be:

`(x-0)^2+(y+7)^2=4^2`

So, the answer will be:

`(x-0)^2+(y+7)^2=16`