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The epsilon-delta definition of a limit is an algebraically precise formulation of evaluating the limit of a function. Informally, the definition states that a limit of a function at a point exists if no matter how is approached, the values returned by the function will always approach . This definition is consistent with methods used to evaluate limits in elementary calculus, but the mathematically rigorous language associated with it appears in higher level analysis. The - definition is also useful when trying to show the continuity of a function.
The equation of any conic can be expressed as
However, the condition for the equation to represent a circle is and . Then the general equation of the circle becomes
Unfortunately, it can be difficult to decipher any meaningful properties about a given circle from its general equation, so completing the square allows quick conversion to the standard form, which contains values for the center and radius of the circle.
Convert into standard form by completing the square.
First, we group the terms together and the terms together while adding 23 to both sides, giving us
When we complete the square for both groups of and terms, we obtain the standard form
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This is the standard equation of a circle, with radius and center at .
The general form can be converted into the standard form by completing the square. First, we combine like terms
Comparing this with the standard form, we come to know that
- center
- radius
However, make sure the coefficients of and are 1 before applying these formulae.
Examples
While solving problem we try to make left hand side of the form by using completing the square method.
Draw the graph of the equation .
Notice that the right side is . Comparing to the standard equation of circle we easily see that the graph is a circle with radius and center at . Now we can easily draw the graph using compass.
What does the graph of the equation look like?
Notice that we can rewrite the equation as
Completing the squares, this becomes
So the graph is a circle with radius , centered at .
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What are the radius and center of the circle whose equation is ?
We can apply completing the square method to left hand side:
Comparing with the standard equation, we can see that and Therefore, the center of the circle is and its radius is
What are the radius and center of the circle whose equation is ?
We can rewrite the given equation as
Comparing with the standard equation, we can see that Therefore the center of the circle is the origin and its radius is !
What is the value of in the figure below?
Figure
Since it is a circle and is touching both the -axis and -axis, its distance from both the axes must be the same. Since it is units away from the -axis, it must be units away from the -axis. Therefore,
Diametric Form
Another way of expressing the equation of a circle is the diametric form.
Suppose there are two points on a circle and , such that they lie on the opposite ends of the same diameter, then the equation of the circle can be written as
Suppose 2 points on the circle and are diametrically opposite, then for any point on the circle, will be a right triangle, with right angle at .
Since can be equal to and ,
Find the equation of the smallest possible circle that passes through the points and
The circle would be the smallest if the two points were to be the endpoints of a diameter of the circle. We can use the diametric form to get