Demonstrate how to find the center and radius of a circle using an equation. Be sure to provide a unique example that shows how to find both of these.

Question

mathstudent55 · Answer

The equation

$(x - h)^2 + (y - k)^2 = r^2$

is the equation of a circle with center $(h ,k)$ and radius $r$.

simplymarie_x · Answer

So is this right?

(x - 1)^2 + (y - 5)^2 = 4 
Center is at (1 , 5). 
Radius is sqrt(4) = 2

simplymarie_x · Answer

@mathstudent55

mathstudent55 · Answer

Correct.

simplymarie_x · Answer

Thanks so much!

mathstudent55 · Answer

Now let's make your problem more interesting.
I will use your equation of a circle.

Let's square both binomials and combine like terms.

$(x - 1)^2 + (y - 5)^2 = 4 $

mathstudent55 · Answer

$(x - 1)^2 + (y - 5)^2 = 4 $

$x^2 - 2x + 1 + y^2 - 10y + 25 = 4$

$x^2 + y^2 - 2x - 10y = -22$

mathstudent55 · Answer

Now say that you problem starts with the last equation above.
You are told that it is the equation of a circle, and your job is to find the center and the radius of the circle.

mathstudent55 · Answer

You take the given equation and you do this:
1. Separate the x-terms and the y-terms on the left side.
2. Complete the square for the x terms and complete the square for the right terms.
3. Factor the x part and the y part, so you have the squares of two binomials.
4. Read the center and the radius from the equation.

mathstudent55 · Answer

Now I'll show you how you actually do it.

simplymarie_x · Answer

You're amazing. I'll all ears :-)

mathstudent55 · Answer

Start with the equation I ended up with.

$x^2 + y^2 - 2x - 10y = -22$

Write the x-terms first followed by the y-terms.

$x^2 - 2x + y^2 - 10y = -22$

Now we need to complete the square twice, once for x and once for y.

$x^2 - 2x + 1 + y^2 - 10y + 25 = -22 + 1 + 25$

Notice that the numbers 1 and 25 were added to the left side to complete the squares.
Those same numbers must be added to the right side, since we must add the same number to both sides of an equation to keep the equation true. This is the "addition property of equality".

mathstudent55 · Answer

Now we rewrite the completed squares as squares of binomials and combine the numbers on the right side.

mathstudent55 · Answer

$(x -1)^2  + (y -5)^2  = 4$

Rewrite the right side as the square of a number to show the radius:

$(x -1)^2  + (y -5)^2  = 2^2$

Now we read: center of the radius: (1, 5); radius of the circle: 2

mathstudent55 · Answer

Thanks, btw.