Question

What is the diameter of a circle given the equation x^2+5x=−8y−y^2

Answer 1

Completing the squares and comparing to the standard equation, it is found that the diameter of the circle is of $\sqrt{89}$ units.

What is the equation of a circle?

• The equation of a circle of center $(x_0, y_0)$ and radius r is given by:

$(x - x_0)^2 + (y - y_0)^2 = r^2$

• The diameter is twice the radius, that is, $d = 2r$.

In this problem, the equation is:

$x^2 + 5x = -8y - y^2$

Then:

$x^2 + 5x + y^2 + 8y$

Completing the squares:

$\left(x + \frac{5}{2}\right)^2 + (y + 4)^2 = \left(\frac{5}{2}\right)^2 + 4^2$

$\left(x + \frac{5}{2}\right)^2 + (y + 4)^2 = \frac{25}{4} + 16$

$\left(x + \frac{5}{2}\right)^2 + (y + 4)^2 = \frac{89}{4}$

Hence:

$r^2 = \frac{89}{4}$

$r = \sqrt{\frac{89}{4}}$

$r = \frac{\sqrt{89}}{2}$

Hence, the diameter is:

$d = 2r = 2\left(\frac{\sqrt{89}}{2}\right) = \sqrt{89}$

You can learn more about the equation of a circle at https://brainly.com/question/16505663