Question

The equation x2 + y2 - 4x + 2y = b describes a circle. The radius of the circle is 7 units. What is the valine of b in the equation?

Answer 1

44 = b

Explanation:

The equation for a circle centered in the point (a, b) and of radius R is:

(x - a)^2 + (y - b)^2 = R^2

Now we start with the equation:

x^2 + y^2 - 4x + 2y = b

First, we can reorder the terms as:

(x^2 + 2*(-2)*x) + (y^2 + 2*y) = b

Now let's complete the squares:

In the first parentheses we can add and subtract 4

(x^2 + 2*(-2)*x + 4 - 4) + (y^2 + 2*y) = b

(x^2 - 2)^2 - 4 + (y^2 + 2*y) = b

Now let's complete the other square, here we need to add and subtract 1.

(x^2 - 2)^2 - 4 + (y^2 + 2*y + 1 - 1) = b

(x^2 - 2)^2 - 4 + (y + 1)^2 - 1 = b

(x^2 - 2)^2 + (y + 1)^2 = b + 1 + 4

Then the radius of this circle is:

R = √(b + 1 + 4)

And we know that R = 7, then:

7 = √(b + 1 + 4)

7^2 = b + 1 + 4

49 = b +5

49 - 5 = b

44 = b