Question

What are the center and the radius of the circle 4x2 + 4y2 - 4x + 16y + 1 = 0?

Answer 1

Answer:

Center of circle: (1, -2)

Radius: √6

Step-by-step explanation:

First, we need to isolate the constant:

Then, we can divide the whole equation by 4:

Then, we can complete the square(reminder: (a+b)^2 = a^2 + 2ab + b^2)

Now that we got the standard equation of the circle, we can find the center and radius by using the equation , where (h, k) is the coordinate of the center, and r is the radius.

Therefore, the center of the circle is (1, -2), and the radius of the circle is √6.

Step-by-step explanation:

Answer 2

The center and radius of the circle 4x² + 4y² - 4x + 16y + 1 = 0 are (1/2, -2) and  2 respectively.

To solve the question, we need to know the standard equation of a circle.

What is the standard equation of a circle?

The standard equation of a circle with center (h,k) and radius, r is

(x - h)² + (y - k)² = r²  (1)

Now, to find the center and radius of 4x² + 4y² - 4x + 16y + 1 = 0, we convert it to standard form by completing the square method.

What is completing the square method?

This is a mathematical operation in which we convert a mathematical expression into a perfect square.

So, 4x² + 4y² - 4x + 16y + 1 = 0

Subtracting 1 from both sides, we have

4x² + 4y² - 4x + 16y + 1 - 1 = 0 - 1

4x² + 4y² - 4x + 16y = - 1

Divding through by 4, we have

x² + y² - x + 4y = -1/4

Adding half the square of the coefficient of x and y to both sides, we have

x² + y² - x + 4y + (-1/2)² + 2² = -1/4 + (-1/2)² + 2²

x² - x + (-1/2)² + y² + 4y + 2² = -1/4 + 1/4 + 2²

(x - 1/2)² + (y + 2)² = 0 + 2²

(x - 1/2)² + (y + 2)² = 2²  (2)

Comparing equations (1) and (2), we have h = 1/2, k = -2 and r = 2.

So, the center and radius of the circle 4x² + 4y² - 4x + 16y + 1 = 0 are (1/2, -2) and  2 respectively.

Learn more about standard equation of a circle here:

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