BuyFind

Precalculus: Mathematics for Calcu...

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

Precalculus: Mathematics for Calcu...

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

Solutions

Chapter 1, Problem 107RE
To determine

To interpret:The given equation x2+y2+2x6y+9=0 and decide if it represents a circle, a point or in the last case if it does not represent any graph.

Expert Solution

Answer to Problem 107RE

The given equation x2+y2+2x6y+9=0 represents a circle with center (1,3) and radius 1 .

Explanation of Solution

Given information:

The equation x2+y2+2x6y+9=0

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.

Moreover, if the center of the circle is nothing but the origin that is (0,0) then the equation reduces to:

  x2+y2=r2

Completing Square method:

To apply this method on an equation of the form: ax2+bx

Method- Make sure that the coefficient of x2=1 and then, add and subtract the square of the half of coefficient of x and then club the terms using formulae:

  (a+b)2=a2+2ab+b2(ab)2=a22ab+b2

Calculation:

Consider the given equation:

  x2+y2+2x6y+9=0

It can also be written as:

  (x2+2x)+(y26y)=9

Now apply Completing Square Method on the two brackets:

  (x2+2x+(22)2(22)2)+(y26y(62)2+(62)2)=9

Now shift the extra terms to the right hand side

  (x2+2x+(22)2)+(y26y+(62)2)=9+(22)2+(62)2

Simplify it to get:

  (x2+2x+12)+(y26y+32)=9+1+9

Recall the identities:

  (a+b)2=a2+2ab+b2(ab)2=a22ab+b2

Use this to get:

  (x+1)2+(y3)2=12

Compare this with the Standard equation of circle which is

  (xh)2+(yk)2=r2

Where (h,k) is the center and r is the radius of the circle.

We get:

  (h,k)=(1,3) And r=1

Thus, the equation x2+y2+2x6y+9=0 represents a circle with center (1,3) and radius 1 .

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