Question

Which of the following equations represents a circle with center (1,-2) and radius 5?

Answer 1

The equation of the circle is $(x - 1)^2 + (y + 2)^2 = 25$. Alternatively, in general form, it is $x^2 + y^2 - 2x + 4y - 20 = 0$.

Answer 2

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

$(x - h)^2 + (y - k)^2 = r^2$

Given:

Center $(h, k) = (1, -2)$

Radius $r = 5$

Substitute the values of $h$, $k$, and $r$ into the standard equation:

$(x - 1)^2 + (y - (-2))^2 = 5^2$

$(x - 1)^2 + (y + 2)^2 = 25$

This is the equation of the circle.

To express it in the general form $x^2 + y^2 + Dx + Ey + F = 0$, we can expand the equation:

$(x^2 - 2x + 1) + (y^2 + 4y + 4) = 25$

$x^2 + y^2 - 2x + 4y + 1 + 4 - 25 = 0$

$x^2 + y^2 - 2x + 4y - 20 = 0$

Both forms are correct representations of the circle.

Explanation of the solution:

The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. Substitute the given center $(1, -2)$ and radius $5$ into this formula to get $(x - 1)^2 + (y - (-2))^2 = 5^2$, which simplifies to $(x - 1)^2 + (y + 2)^2 = 25$. This equation can also be expanded to $x^2 + y^2 - 2x + 4y - 20 = 0$.