# The equation for the given circle which has points P ( 2 , 3 ) and Q ( − 1 , 8 ) . Also, its center is the midpoint of the given segment P Q ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1, Problem 106RE
To determine

## To calculate:The equation for the given circle which has points P(2,3) and Q(−1,8) . Also, its center is the midpoint of the given segment PQ

Expert Solution

The equation for the circle thathas points P(2,3) and Q(1,8) , center is the midpoint of the given segment PQ is x2+y2x11y+22=0

### Explanation of Solution

Given information:

A circle withpoints P(2,3) and Q(1,8) lying inside it and whose center is given by midpoint of PQ

Formula used:

For a given circle with center (h,k) and radius r the equation is given as:

(xh)2+(yk)2=r2

This is referred to as the Standard form for the equation of a given circle.

The Distance denoted by say, D between any two given points say (x1,y1) and (x2,y2) is given as:

D=(x2x1)2+(y2y1)2

The midpoint (x,y) for two points (x1,y1) and (x2,y2) is given as:

(x,y)=(x1+x22,y1+y22)

The relation between d , diameter of circle and r , radius of circle is:

r=d2

Calculation:

Consider the given circle with points P(2,3) and Q(1,8) lying inside it.

Then in order to apply the midpoint formula for the segment PQ the following should be known:

x1=2x2=1y1=3y2=8

Recall the midpoint (x,y) for two points (x1,y1) and (x2,y2) is given as:

(x,y)=(x1+x22,y1+y22)

Therefore,

(x,y)=(2+(1)2,3+82)=(12,112)

Hence, center of the circle is (x,y)=(12,112)

Now as midpoint of PQ forms the center of the circle, therefore length of PQ gives the diameter for the circle.

And the relation between d , diameter of circle and r , radius of circle is:

r=d2

Therefore, distance from center (x,y)=(12,112) to P(2,3) gives the radius for the circle.

Recall the distance D between two points (x1,y1) and (x2,y2) is given as:

D=(x2x1)2+(y2y1)2

As we have:

x1=12x2=2y1=112y2=3

Therefore,

D=(212)2+(3112)2=(32)2+(52)2=94+254=344=342

Also we have center (h,k)=(12,112)

Recall that for a given circle with center (h,k) and radius r the equation is given as:

(xh)2+(yk)2=r2

Therefore, replacing the values we get:

(x12)2+(y112)2=(342)2 (1)

Now using the following formulae:

(a+b)2=a2+2ab+b2(ab)2=a22ab+b2

Then, (1) reduces to:

(x12)2+(y112)2=(342)2x2+14x+y2+121411y=344x2+y2x11y+(14+1214344)=0x2+y2x11y+(884)=0x2+y2x11y+22=0

Hence, the equation for the circle that has points P(2,3) and Q(1,8) , center is the midpoint of the given segment PQ is x2+y2x11y+22=0

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