Question

Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit.
x2 + 2x + y2 + 4y = 20

Answer 1

Answer:

center: (-1, -2)

radius: 5

Step-by-step explanation:

in picture

Answer 2

Answer:

The circle is of radius $5$ and is centered at $(-1,\, -2)$.

Step-by-step explanation:

Equation for a circle of radius $r$ ($r > 0$) and center $(a,\, b)$:

$(x - a)^{2} + (y - b)^{2} = r^2$.

Expand and rearrange to obtain an equivalent equation:

$\left(x^2 - 2\, a\, x + a^{2}\right) + \left(y^2 - 2\, b\, y + b^2\right) = r^{2}$.

$x^2+ (-2\, a)\, x + y^2 + (-2\, b)\, y = r^2 - (a^2 + b^2)$.

Compare this equation with the equation of the circle in this question:

• Coefficient of the $x$ term: $2$ in the original equation, $(-2\, a)$ in the proposed equation.
• Coefficient of the $y$ term: $4$ in the original equation, $(-2\, b)$ in the proposed equation.
• Constant term: $20$ in the original equation, $\left(r^2 - \left(a^2 + b^2 \right)\right)$ in the proposed equation.

Assume that the proposed equation matches the equation from the question. Each of the coefficients should match.

From the coefficient of the $x$ term and the $y$ term: $a = -1$ and $b = -2$.

Hence, the center of this circle is at $(a,\, b) = (-1,\, -2)$.

The constant term of the two equations should also match: $\left(r^2 - \left(a^2 + b^2 \right)\right) = 20$.

$\left(r^2 - \left((-1)^2 + (-2)^2 \right)\right) = 20$.

$r^2 = 25$.

$r = 5$ (given that radius $r > 0$.)

Hence, the radius of this circle would be $5$.