Question

21. The equation x² + y² - 6x + 8y = 74

represents circle C.
What is the radius of circle C?

Answer 1

7

Explanation:

The best way in which to approach solution of this problem is to use "completing the square." x^2 - 6x becomes x^2 - 6x + 9 - 9 and y^2 + 8y becomes y^2 + 8y + 16 - 16.

Thus, the original equation becomes:

x^2 - 6x + 9 - 9 + y^2 + 8y + 16 - 16 = 74

It's best to rewrite x^2 - 6x + 9 as (x - 3)^2 and y^2 + 8y + 16 as (y + 4)^2 now:

(x - 3)^2 + (y + 4)^2 - 9 - 16 = 74

Consolidating all the constants on the right side, we get:

(x - 3)^2 + (y + 4)^2 = 49

Compare this to the standard equation of a circle with center at (h, k) and radius r:

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

Thus, h must be 3, k must be -4 and r must be sqrt(49), or 7.

The radius of circle C is 7.

Answer 2

The radius is
`3√(11)`.

Explanation:

The standard equation for a circle is
`(x-h)^2+(y-k)^2=r^2`.

I like to call this form center-radius form because it tells you the center (h,k) and the radius r.

So we put it in this form to determine r, the radius, which is the desired value here.

We will need to complete the square.

Thankfully there is a formula to take more thinking out of the process:

`x^2+dx+((d)/(2))^2=(x+(d)/(2))^2`.

Let's begin:

`x^2+y^2-6x+8y=74`

Reorganize the equation so the x's are together and the y's are together.

`x^2-6x+y^2+8y=74`

Now time to being completing the square. Keep in mind whatever you add to one side you need to add the other side to keep the equation equivalent to what it once was.

`x^2-6x+((-6)/(2))^2+y^2+8y+((8)/(2))^2=74+((-6)/(2))^2+((8)/(2))^2`

Use the other side of the formula I mentioned for completing the square:

`(x+(-6)/(2))^2+(y+(8)/(2))^2=74+(-3)^2+(4)^2`

`(x-3)^2+(y+4)^2=74+9+16`

`(x-3)^2+(y+4)^2=74+25`

`(x-3)^2+(y+4)^2=99`

So the center is (3,-4).

The radius is
`√(99)`.

You can actually simplify the radius since 99 contains a factor that is a perfect square.

`√(99)=√(9)√(11)=3√(11)`