# To interpret: The given equation 2 x 2 + 2 y 2 − 2 x + 8 y = 1 2 and decide if it represents a circle, a point or in the last case if it does not represent any graph. ### 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 108RE
To determine

## To interpret:The given equation 2x2+2y2−2x+8y=12 and decide if it represents a circle, a point or in the last case if it does not represent any graph.

Expert Solution

The equation 2x2+2y22x+8y=12 represents a circle with center (12,2) and radius 32 .

### Explanation of Solution

Given information:

The equation 2x2+2y22x+8y=12

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:

2x2+2y22x+8y=12

It can also be written as:

(2x22x)+(2y2+8y)=12

Dividing throughout by 2 to make coefficients of x2 and y2 as 1 to get:

(x2x)+(y2+4y)=14

Now apply Completing Square Method on the two brackets:

(x2x+(12)2(12)2)+(y2+4y+(42)2(42)2)=14

Now shift the extra terms to the right hand side

(x2x+(12)2)+(y2+4y+(42)2)=14+(12)2+(42)2

Recall the identities:

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

Use this to get:

(x12)2+(y+42)2=14+14+4(x12)2+(y+2)2=(32)2

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)=(12,2) And r=32

Thus, equation 2x2+2y22x+8y=12 represents a circle with center (12,2) and radius 32 .

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