Consider the circle of equation x 2 + 2x + y 2 − 8y = 19. (a) Find the center and radius of the circle. (b) Find an equation of the form y = f(x) to describe (i) the top semi-circle and (ii) the bottom semi-circle. (c) Find an equation of the form x = g(y) to describe (i) the left semi-circle and (ii) the right semi-circle.

You want the center and radius of the circle with equation x² +2x +y² -8y = 19, and you want equations for the top and bottom halves, as well as the left and right halves.

(a) Standard form equation

The equation can be written in standard form by "completing the square" for each variable. That is, we will add the square of half the coefficients of the linear terms.

(x² +2x) +(y² -8y) = 19

(x² +2x +1) +(y² -8y +16) = 19 +1 +16

(x +1)² +(y -4)² = 36

Comparing this to the generic equation for a circle with center (h, k) and radius r, we see ...

(x -h)² +(y -k)² = r²

(h, k) = (-1, 4) and r = 6

The center of the circle is (-1, 4) and the radius is 6.

(b) y = f(x)

Solving the standard form equation for y, we find ...

(y -4)² = 36 -(x +1)²

y -4 = ±√(36 -(x +1)²)

y = ±√(36 -(x +1)²) +4 . . . . . . . top and bottom halves

The top half equation is ...

y = √(36 -(x +1)²) +4

The bottom half equation is ...

y = -√(36 -(x +1)²) +4

(c) x = f(y)

Similarly, solving for x gives ...

x = ±√(36 -(y -4)²) -1

The equation for the left half is ...

x = -√(36 -(y -4)²) -1

The equation for the right half is ...

x = √(36 -(y -4)²) -1

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Additional comment

This particular graphing program chokes on the equations for x=f(y), so doesn't graph them properly. Hence, we are only showing the equation for the full circle.