Q. 33 - Precalculus Enhanced with Graphing Utilities

Question

Find (a)  the center (h, k) and radius r of each circle; (b) graph each circle; (c) find the intercepts if any of the equation $2x^2+8x+2y^2=0$

Step-by-Step Solution

Answer

a) center (h ,k), and the radius of the given equation  are (-2,0) and 2

b) graph of the given equation is given below

Part (a) step 1. Given information

Given the equation of the circle is $2x^2+8x+2y^2=0$

We have to   Find (a)  the center (h, k) and radius r of  circle

Part (a) Step 2. Rewriting the equation of a circle

Here equation is in the form$$\begin{gathered} 2x^2+8x+2y^2=0 \\ {x}^2+4x+y^2=0 dividing by 2 get equation in s\tan dard form \\ (x^2+4x)+y^2=0 \end{gathered}$$

Part (a) step 3. Find center and radius

complete the square of each expression in parentheses, we get an

$$\begin{gathered} (x^2+4x)+y^2=0 \\ (x^2+4x+4)+y^2=4 ,\sin ce any number added on the left side must add to right side \\ {(x+2)}^2+{(y-0)}^2=2^2 so this s\tan dard form of the circle equaion with center and radius \\ hence center is(-2,0) and radius is 2. \end{gathered}$$

 Here x intercept and y intercept are both at 0.

Part (b) step 1. Given information

 Here given equation of a circle is$2x^2+8x+2y^2=0$

We need to sketch the graph

Part (b) step 2. Graph of the equation of the circle

   the graph of the given equation with center(-2,0) and radius 2 is given below