Points O (0, 0) and B (0, 3) below lie in the standard (x, y) coordinate plane. The collection of all points such that each is twice as far from B as from O forms a circle. The point (3‾√,0) is one point on the circle. What are the coordinates of the center of that circle?

The coordinates of the centre of our circle = (0, -1)

Step-by-step explanation:

Let (x,y) represent a point on the circle.

Distance between (x,y) and B(0,3) is 2 × distance between (x,y) and O(0,0)

Distance between two points (x₁, y₁) and (x₂, y₂) in coordinates system is given as

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

So, the statement from the question can be interpreted as

√[(x - 0)² + (y - 3)²] = 2 × √[(x - 0)² + (y - 0)²]

Squaring both sides

x² + (y - 3)² = 4(x² + y²)

4x² + 4y² = x² + y² - 6y + 9

3x² + 3y² + 6y - 9 = 0

Divide through by 3

x² + y² + 2y - 3 = 0

x² + y² + 2y = 3

Add 1 to both sides

x² + y² + 2y + 1 = 3 + 1

x² + y² + y + y + 1 = 4

x² + (y + 1)² = 2²

(x + 0)² + (y + 1)² = 2²

Comparing this to the equation of a circle

(x - 0)² + (y - b)² = r²

where r = radius of the circle, and the coordinates of the circle's centre is (a,b)

[x - 0]² + [y - (-1)]² = 2²

The centre of our circle = (0, -1) and radius is 2.

To check, let's see if point (√3, 0) is indeed a point on the circle. x = √3 and y = 0

x² + (y + 1)² = 2²

(√3)² + (0 + 1)² = 2²

3 + 1 = 4

4 = 4

Hence, our equation of the circle is correct and the point (√3, 0) is truly on the circle.

Hope this Helps!!!

ratio is 2 : 9

the answer is three

13.5 : 3