In this section we will explore how we determine from the equation whether we are looking at an equation of a circle, parabola, ellipse or hyperbola. The first thing we look at is are both x and y squared or is just one variable squared. If only one is squared then we know it is a parabola and we proceed with getting the equation in standard form for a parabola so we can find the appropriate information needed.

If both variables are squared and there is a subtraction sign between them, then we are dealing with a hyperbola and we proceed with getting the equation in standard form for a hyperbola so we can find the appropriate information needed.

If both variables are squared and there is an addition sign between them, it is an ellipse if the coefficients ( the numbers in front of the x and y squared) are different and a circle if they are the same. We then proceed accordingly to get them into their standard forms to find the appropriate information needed.

So lets start with the example:

4x² + 4y² - 8x -8y + 7 = 0

Can you tell which conic section is described by this equation? Check below to see if you are correct.

If you said circle you are correct because the coefficients are equal and there is an addition sign between the two squared terms. Now do you remember how to get a circle equation into standard form? You must complete the square. Try it and then check your answer below.

The first thing we need to do is group the x terms together and leave a plus "blank" and then group the y terms together and leave a plus "blank", take the 7 to the other side by subtracting it and then leave two plus "blanks" so that we can add something in each blank and keep our equation balanced.

So it should look like this so far:

4x² - 8x + ___ + 4y² - 8y + ____= -7 + ____+ _____

Now do you remember what else you need to do before you can fill in the blank? Check below.

If you said factor the four out of each term, you are correct. Your equation should now look like this:

4(x² - 2x + ___) + 4(y² - 2y + ____) = -7 + ____+ _____

Now we fill in the blank on the left side by taking the coefficient of the middle term, dividing it by -2 and squaring it...so we take -2/2 which is -1 and square it which is 1 and place it in the first blank. We do the same with the middle term of the y's....we take -2/2 and square it so we get 1 again and place it in the second blank. To fill in the blanks on the right side of the equation we must multiply the number we put in the blank by the number on the outside of the parentheses and then put them each in the blanks on the right side so try it and check it below.

This is what your equation should look like right now:

4(x² - 2x + 1) + 4(y² - 2y + 1) = -7 + 4 + 4

Now we factor the left and simplify the right and divide by 4 so that we can find the center and radius. Try it and check down below.

Our final equation should look like this:

(x -1)² + (y - 1)² = 1/4

(remember you divided by 4, so after you add -7, 4 and 4 you must divide the right side by 4 also.)

Therefore the center is (1, 1) (remember we find the center by taking the opposite of each number being subtracted from the x and y variables) and the radius in this case is 1/2 because you take the square root of the number on the right side. You can then graph it by graphing the center and counting 1/2 a unit in each direction from the center. Look down below for the graph of the circle.

This is the graph of the circle:

Lets try another equation. Suppose we start with the equation:

9x² + 25y² + 36x -150y + 36 = 0

Can you tell which conic section this represents? Check below.

If you said ellipse, you are correct because the two variables are both squared with an addition sign between them and different numbers as coefficients in front of the squared terms. So now, we need to complete the square again. Try on your own and if you get stuck, check below to see what happened.

9x² + 36x + ____ + 25y² - 150y + ____ = -36 + ____ + ____

9(x² + 4x + ____) + 25(y² - 6y + ____) = -36 + ____ + ____(factor out the coefficient)

9(x² + 4x + 4) + 25(y² - 6y + 9) = -36 +36 + 225 (take half of middle term square)

9(x + 2) ² + 25(y - 3)² = 225 (factor left and simplify right)

Divide by 225 to get a 1 on the right side:

(x + 2)²     (y - 3)²
----------- + ---------- = 1
   25            9

Now we are ready to name the center and find the foci. Can you guess what the center is? Check below.

The center is (-2, 3), but now we must find the foci. To do this we must first find c. It is going to be a little more complicated than before since the center is not (0,0). Can you try to find c from what you learned before? Check below.

To find c, we use the equation: b² = a² - c². From the equation above we know a² = 25 and b²= 16...so we get 16 = 25 - c² . Solving for c, we get c² = 9, c = 3. Now we must use the fact that the major axis is the x axis since the largest number is under the x variable....so we must subtract our c value from the x coordinate of our center and add it to our x coordinate of the center so we get a focus and both sides of the center where the foci are located. If the y axis was the major axis we would add and subtract the c value from the y coordinate of the center. Try writing the foci and check below to see if it is correct.

The foci are: ( -2 + 3, 3) and (-2 - 3, 3) giving us the points...(1, 3) and (-5, 3). We now use all this information to graph our ellipse. Try it and then check below.

The graph of the ellipse should look like this:

Let's explore another example. Suppose we have the equation:

      4x² - 25y² -24x +50y -89 = 0

Can you tell me the type of conic this is? Look below to see if you are correct.

This is a hyperbola because there are two squared variables and there is a subtraction sign between them. So now we again complete the square to put the equation into standard form. Try to do the work then follow the steps below to see if you are correct.

4x² - 24x +____ - 25y² + 50x + ____ = 89 +____+____ (group like variables)
4( x² - 6x + ___) - 25(y² - 2x + ___) = 89 +____+____ (factor out the coefficients)
4(x² - 6x + 9) - 25(y² - 2x + 1) = 89 + 36 - 25 (divide middle term by 2 and square)
4(x - 3)² - 25(y - 1)² = 100 (factor left side and simplify right side)

Divide by 100: 

(x -3)²          (y-1)²
---------    -     -------    =    1
  25                4

Now can you find the center and a², b² and c² so that you can find the foci and graph it? Look below to check.

The center should be (3,1). a² = 25 because it is before the subtraction sign. b² = 4. So now you substitute these values into the equation b² = c² - a ² to obtain 4 = c² - 25, giving us c = √29.

Since the major axis is the x because it is before the subtraction sign....we add and subtract √29 to the x coordinate of the center to get the foci which are located on either side of the center on the major axis. Therefore the coordinates of the foci are:
(3 + √29, 1) and (3 - √29, 1). We then take all this information and graph it. Check your answer below.

Here is the graph of the hyperbola:

The last example we will look at here is : x² - 6x -4y +5 =0. From the descriptions at the top of the page....can you tell which type of conic this is? Check below.

If you said parabola, you are correct. There is only one variable squared this time which is what makes this equation a parabola. Now we must rewrite this equation by completing the square to put it into standard form to find the vertex, focus, directrix and axis of symmetry. Try to complete the square and check your answer below.

x² - 6x + ___ = 4y - 5 + ___(group variable squared,take rest to other side of equation)
4y - 5 + ___ = x² - 6x + ___ (reverse the equation so squared term is on right side)
4y - 5 + 9 = x² - 6x + 9 (fill in blank with middle term divided by 2 and squared)
4y + 4 = (x - 3)² (simplify left side and factor right side)
y + 1 = (1/4)(x - 3)² (divide by coefficient in front of unsquared variable, make it a fraction on the right side in front of the parantheses.)

Now we should be able to find the 4 things above...the vertex, focus, directrix and axis of symmetry. See what you can do and then check below.

The vertex of this parabola is ( 3, -1). Because the x is squared, this parabola opens up or down. Then when we look at the number in front of the parantheses and see that this is positive we can tell that the parabola opens up.

Also the number in front of the paranthese represents a which equals (1/4c). Setting (1/4) = to (1/4c) and solving for c, we get c = to 1. This gives us the distance from the vertex to the focus and from the vertex to the directrix because these distances are equal.

We now use 1 and add it to the y coordinate of the vertex to get the focus of (3, 0) and then we subtract it from the y coordinate of the vertex to get the directrix of: y = - 2. The axis of symmetry is the line that cuts the parabola in half and goes through the vertex, so it is the equation: x = 3.

Now we are ready to graph all of this. Try graphing it and then check your answer below.

Here is the graph of the parabola with the 4 pieces of information labeled.