Question

1. If the Center of a circle is at (2, -7) and a point on the circle (5,6) find the formula of the circle.

2. What surfaces in R3 are represented by the following equations?
z = 3
y = 5


3. Find an equation of a sphere with radius r and center C(h, k, l).

4. Show that x2 + y2 + z2 + 4x – 6y + 2z + 6 = 0 is the equation of a sphere. Also, find its center and radius.

Answer 1

1.The equation of the circle

$$(x-x_C)^2+(y-y_C)^2=r^2$$

$x_C=2, y_C=-7$

Point $(5,6)$ lies on the circle

$$(5-2)^2+(6+7)^2=r^2$$

$$r^2=178$$

The equation of the circle

$$(x-2)^2+(y+7)^2=178$$

2. In three-space, the equation $z=3$ represents a horizontal plane, parallel to the xy-plane and at height 3. 

In three-space, the equation $y=5$ represents a vertical plane parallel to the xz-plane and passing through the point $(0,5,0)$.

3. The general equation of a sphere is:

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

where $(h,k,l)$ represents the center $C$ of the sphere, $r$ represents the radius, and $x,y,$ and $z$ are the coordinates of the points on the surface of the sphere.

4.

$$x^2+y^2+z^2+4x-6y+2z+6=0$$

$$x^2+4x+4-4+y^2-6y+9-9+z^2+2z+1-1+6=0$$

$$(x+2)^2+(y-3)^2+(z+1)^2=8$$

This is the equation of a sphere with radius $r=2\sqrt{2}$ and center $C(-2, 3, -1).$