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 109RE
To determine

To interpret:The given equation x2+y2+72=12x 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 109RE

The equation x2+y2+72=12x does not represent any graph.

Explanation of Solution

Given information:

The equation x2+y2+72=12x

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.

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+72=12x

It can also be written as:

  (x212x)+(y0)2=72

Now apply Completing Square Method on the first bracket:

  (x212x+(122)2)(122)2+(y0)2=72

Now shift the extra terms to the right hand side

  (x212x+(6)2)+(y0)2=72+(6)2(x212x+(6)2)+(y0)2=72+36

Recall the identities:

  (a+b)2=a2+2ab+b2(ab)2=a22ab+b2

Use this to get:

  (x6)2+(y0)2=36

Which is not possible, as sum of two positive quantities cannot yield a negative quantity.

Thus, the equation x2+y2+72=12x does not represent any graph.

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