We can write the boundary curve for Inequality (I) by replacing the less than sign with an equals sign. Then, we can identify which conic section it is.
x^2+y^2=36
Let's recall the standard equation of a circle.
(x- h)^2+(y- k)^2= r^2
Here, the center is the point ( h, k) and the radius is r. We will rewrite the given equation to match this form, and then we can identify the center and the radius.
x^2+y^2=36 [0.3em] ⇕ [0.3em] (x- 0)^2+(y- 0)^2= 6^2
The center of the circle is the point ( 0, 0), and its radius is 6. Let's draw its graph.
Now that we have the boundary curve, we need to determine which region to shade. To do so, we will use (0,0) as a test point. If the point satisfies the inequality, we will shade the region that contains the point. If not, we will shade the opposite region.
Since the substitution produced a true statement, we will shade the region that contains the point (0,0). Because we have a strict inequality, the boundary curve will be dashed.
Inequality (II)
We can write the boundary curve by replacing the greater than sign with an equals sign.
4x^2+9y^2=36
To identify which conic section it is we need to rewrite the equation.
Notice that the denominator of the expression containing the x-variable is greater than the denominator of the expression that contains the y-variable. Therefore, we have a horizontalellipse. Let's recall the main characteristics of this type of ellipse.
Horizontal Ellipse
Standard-Form Equation
(x- h)^2/a^2+(y-k)^2/b^2=1
Center
( h,k)
Vertices
( h± a,k)
Co-vertices
( h,k ± b)
Foci
( h± c,k)
a,b,c relationship, a>b>0
c^2= a^2- b^2
Let's consider our equation one more time.
(x- 0)^2/3^2+(y- )^2/2^2=1
We can see that a= 3, b= 2, h= 0, and k= . The only value we do not know is c. Let's find it!
Let's now draw the ellipse on the same set of axis as we drew the circle. Because the inequality is strict, the boundary curve will be dashed.
Now that we have the boundary curve, we need to determine which region to shade. Again, we will use (0,0) as a test point. Let's determine if the point satisfies the inequality or not.
