Q 63 - Precalculus Enhanced with Graphing Utilities

Question

Open the “Circle: the role of the center” applet. Place the cursor on the center of the circle and hold the mouse button. Drag the center around the Cartesian plane and note how the equation of the circle changes.

(a) What is the radius of the circle?

(b) Draw a circle whose center is at (1, 3). What is the equation of the circle?

(c) Draw a circle whose center is at ( -1, 3). What is the equation of the circle?

(d) Draw a circle whose center is at (-1, -3). What is the equation of the circle?

(e) Draw a circle whose center is at (1, -3). What is the equation of the circle?

(f) Write a few sentences explaining the role the center of the circle plays in the equation of the circle.

Step-by-Step Solution

Answer

(a) The radius of the circle is the length of the segment from the center to any point on its edge. center.

(b) The equation of the circle is $(x-1{)}^2+(y-3{)}^2=r^2$

(c)  The equation of the circle is  $(x+1{)}^2+(y-3{)}^2=r^2$

(d)  The equation of the circle is $(x+1{)}^2+(y+3{)}^2=r^2$

(e) The equation of the circle is $(x-1{)}^2+(y+3{)}^2=r^2$

(f) The center of the circle plays a vital role in the equation of the circle as it specifies the coordinate in which the circle is lying.

Step 1. Given information

Open the “Circle: the role of the center” applet. Place the cursor on the center of the circle and hold the mouse button. Drag the center around the Cartesian plane and note how the equation of the circle changes.

Part (a) of Step 1. Radius of the circle

The radius of the circle is the length of the segment from the center to any point on its edge. center. 

Part (b) of Step 1. Equation of the circle

The given center of the circle $\left(1, 3\right)$

The standard equation of the circle is $(x-h{)}^2+(y-k{)}^2=r^2$

Identify h and k from the center of the circle, $\left(h,k\right)=\left(1,3\right)$

Substitute the value of h and k in the equation of the circle,

$(x-1{)}^2+(y-3{)}^2=r^2$

The required circle with center (1,3) is 

Part (c) of Step 1. Equation of the circle

The given center of the circle $(-1,3)$

The standard equation of the circle is $(x-h{)}^2+(y-k{)}^2=r^2$

Identify h and k from the center of the circle, $\left(h,k\right)=\left(-1,3\right)$

Substitute the value of h and k in the equation of the circle, 

$(x+1{)}^2+(y-3{)}^2=r^2$

The required circle with a center (-1,3) is 

Part (d) of Step 1. Equation of the circle

The given center of the circle $\left(-1,-3\right)$

The standard equation of the circle is $(x-h{)}^2+(y-k{)}^2=r^2$

Identify h and k from the center of the circle, $\left(h,k\right)=\left(-1,-3\right)$

Substitute the value of h and k in the equation of the circle

$(x+1{)}^2+(y+3{)}^2=r^2$

The required circle with a center (-1,-3) is 

Part (e) of Step 1. Equation of the circle

The given center of the circle $\left(1,-3\right)$

The standard equation of the circle is $(x-h{)}^2+(y-k{)}^2=r^2$

Identify h and k from the center of the circle, $\left(h,k\right)=\left(1,-3\right)$

Substitute the value of h and k in the equation of the circle

$(x-1{)}^2+(y+3{)}^2=r^2$

The required circle with a center (1,-3) is 

Part (f) of Step 1. Role the center of the circle plays in the equation of the circle.

The center of the circle plays a vital role in the equation of the circle as it specifies the coordinate in which the circle is lying.