# To interpret: The given equation x 2 + y 2 − 6 x − 10 y + 34 = 0 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 110RE
To determine

## To interpret:The given equation x2+y2−6x−10y+34=0 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 x2+y26x10y+34=0 represents a point (3,5) .

### Explanation of Solution

Given information:

The equation x2+y26x10y+34=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+y26x10y+34=0

It can also be written as:

(x26x)+(y210y)=34

Now apply Completing Square Method on the two brackets:

(x26x+(62)2(62)2)+(y210y+(102)2(102)2)=34

Now shift the extra terms to the right hand side

(x26x+(62)2)+(y210y+(102)2)=34+(62)2+(102)2

Recall the identities:

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

Use this to get:

(x26x+32)+(y210y+52)=34+32+52(x3)2+(y5)2=0

Now sum of two positive quantities can be 0 only when each of them is individually equal to 0 .

Therefore we get:

(x3)2=0x3=0x=3

And

(y5)2=0y5=0y=5

Thus,the equation x2+y26x10y+34=0 represents a point (3,5) .

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