Chapter 1, Problem 109RE

To determine

To interpret:The given equation $x^2+y^2+72=12x$ and decide if it represents a circle, a point or in the last case if it does not represent any graph.

Answer to Problem 109RE

The equation $x^2+y^2+72=12x$ does not represent any graph.

Explanation of Solution

Given information:

The equation $x^2+y^2+72=12x$

Formula used:

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

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

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: $a{x}^2+bx$

Method- Make sure that the coefficient of $x^2=1$ and then, add and subtract the square of the half of coefficient of $x$ and then club the terms using formulae:

  $\begin{array}{l}{\left(a+b\right)}^2=a^2+2ab+b^2\\ {\left(a-b\right)}^2=a^2-2ab+b^2\end{array}$

Calculation:

Consider the given equation:

  $x^2+y^2+72=12x$

It can also be written as:

  $\left(x^2-12x\right)+{\left(y-0\right)}^2=-72$

Now apply Completing Square Method on the first bracket:

  $\left(x^2-12x+{\left(\frac{-12}{2}\right)}^2\right)-{\left(\frac{-12}{2}\right)}^2+{\left(y-0\right)}^2=-72$

Now shift the extra terms to the right hand side

  $\begin{array}{l}\left(x^2-12x+{\left(-6\right)}^2\right)+{\left(y-0\right)}^2=-72+{\left(-6\right)}^2\\ \left(x^2-12x+{\left(-6\right)}^2\right)+{\left(y-0\right)}^2=-72+36\end{array}$

Recall the identities:

  $\begin{array}{l}{\left(a+b\right)}^2=a^2+2ab+b^2\\ {\left(a-b\right)}^2=a^2-2ab+b^2\end{array}$

Use this to get:

  ${\left(x-6\right)}^2+{\left(y-0\right)}^2=-36$

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

Thus, the equation $x^2+y^2+72=12x$ does not represent any graph.