Q.43 - Precalculus Enhanced with Graphing Utilities

Question

match each graph with the correct equation.$$\begin{gathered} \left(a\right) {(x-3)}^2+{(y+3)}^2= 9, \left(b\right) {(x+1)}^2+{(y-2)}^2=4 , \\ \left(c\right) {(x-1)}^2+{(y+2)}^2=4 , \left(d\right) {(x+3)}^2+{(y-3)}^2=9 \end{gathered}$$

Step-by-Step Solution

Answer

The correct equation of the given graph is option $\left(c\right) {(x-1)}^2+{(y+2)}^2=4$

Step 1. Given information

Here graph of the circle is given.

 we need to match the graph with the correct equation. 

Step 2 . Plot the graph of option ( a )   ( x - 3 ) 2 + ( y - 3 ) 2 = 9

Here given equation is  ${(x-3)}^2+{(y+3)}^2=9$  which is in the standard form of the equation of a circle ${(x-h)}^2+{(y-k)}^2=r^2 , here( h, k) is center of the circle and radius is r$. Thus when we compare the given equation to a standard form of the equation the circle, we get center  (h, k) is (3,-3) and radius is 3. Then  the graph of the given equation is given below

Step 3. Plot the graph of option ( b )   ( x + 1 ) 2 + ( y - 2 ) 2 = 4  

Here  equation of the circle ${(x+1)}^2+{(y-2)}^2=4$, when we compared this to a standard form of the equation of a circle, we can understand that center (h, k) is  ((-1,2) and radius is 2. then graph is given here

Step 4. Plot the graph of option ( c )   ( x - 1 ) 2 + ( y + 2 ) 2 = 4

Here equation of the equation of a circle is ${(x-1)}^2+{(y+2)}^2=4$ . From the given equation we get the center (h, k)  is (1.-2 and radius is 2 since the given equation is in the form of a standard equation of a circle${\left(x-h\right)}^2+{(y-k)}^2=r^2$. Hence the graph is given below

Step 5. Sketch the graph of option   ( d )   ( x + 3 ) 2 + ( y - 3 ) 2 = 9

To plot the graph of the circle we should know the center and radius , here the equation is in the form of the standard form of the equation of a circle ${(x-h)}^2+{(y-k)}^2=r^2$.  Thus the center of the circle according to the question is (-3,3) and the radius is 3. Then graph is given below

Step 6. Matching the correct equation

According  to the question,  we plotted the graph of the equation for each option and the correct match of equation of  the given question graph is option (c) ${(x-1)}^2+{(y+2)}^2=4$