Chapter 12.5, Problem 50SR

To determine

To find: Whether the graph of each equation is parabola, circle, ellipse, or hyperbola.

Answer to Problem 50SR

Graph of given equation is Circle

Explanation of Solution

Given information:

  $x^2+y^2-8x-6y+5=0$

Concept:

Equation in the form

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Then we need to calculate discriminant

  $\Delta =B^2-4AC$

If $\Delta =0$ and $A=0$ or $C=0$

Then the figure is Parabola

If

  $\Delta >0$

Then the figure is Hyperbola

If $\Delta <0$ and $A=C$

Then the figure is Circle.

If $\Delta <0$ and $A\ne C$

Then the figure is Ellipse.

Given

  $x^2+y^2-8x-6y+5=0$

On comparing with standard equation

We get

  $\begin{array}{l}A=1\\ B=0\\ C=1\\ D=-8\\ E=-6\\ F=5\end{array}$

Substituting values in discriminant formula

We get

  $\Rightarrow \Delta =0^2-4\left(1\right)\left(1\right)=-4<0$

Here

  $\Delta <0$ and $A=C$

So, given equation is Circle.