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Math  /  Geometry

QuestionWrite the equation of this circle in standard form. □\square □\square 2

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a circle drawn on a graph, and write it in the standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2. Watch out! Don't mix up the coordinates of the center (h,k)(h, k) and the radius rr!

STEP 2

1. Find the Center
2. Find the Radius
3. Write the Equation

STEP 3

Alright, let's **locate the center** of this awesome circle!
Looking closely at the graph, we can see that the center of the circle is at the point (0,6)(0, 6).
So, our **center** is (h,k)=(0,6)(h, k) = (0, 6).

STEP 4

Now, let's **find the radius**.
The radius is the distance from the center to any point on the circle.
We can easily see that the circle goes from y=3y=3 to y=9y=9, which means the **diameter** is 9−3=69-3=6.
Since the radius is half of the diameter, our **radius** is r=62=3r = \frac{6}{2} = 3.

STEP 5

Time to **plug in** our values into the **standard form** equation of a circle: (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2.

STEP 6

We found that the **center** is (0,6)(0, 6), so h=0h = 0 and k=6k = 6.
We also found that the **radius** is 33, so r=3r = 3.

STEP 7

Substituting these values into the equation, we get (x−0)2+(y−6)2=32(x-0)^2 + (y-6)^2 = 3^2.

STEP 8

Simplifying, we get x2+(y−6)2=9x^2 + (y-6)^2 = 9.
Boom!

STEP 9

The equation of the circle in standard form is x2+(y−6)2=9x^2 + (y-6)^2 = 9.

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