Introduction to Conics

Conics Review

Circles, Ellipses, Hyperbolas and Parabolas

Circles

• The circle equation requires us to know 2 things:

• The center (h,k)

• The radius R

• If we know the center and radius, the equation is:  $\left(x-h\right)^2+\left(y-k\right)^2=R^2$  

Multiple Select

Which of the following are circles?

• $$\left(x-2\right)^2+\left(y-1\right)^2=1$$

• $$x^2+y^2=16$$

• $$\left(x-4\right)^2-\left(y+3\right)^2=1$$

• $$\frac{\left(x-5\right)^2}{4}+\frac{\left(y+2\right)^2}{4}=1$$

Multiple Choice

$$\left(x+1\right)^2+\left(y-5\right)^2=25$$  What are the center and radius of the following circle?

• center = (1, -5)
  radius = 25

• center = (-1, 5)
  radius = 25

• center = (1, 25)
  radius = 25

• center = (-1, 5)
  radius = 5

Multiple Choice

What are the center and radius of the following circle?

• center = (-2, 2)

  radius = 4

• center = (-2, -2)

  radius = 2

• center = (-2, 2)

  radius = 2

Multiple Choice

What is the equation of the following circle?

• 
  

  $$\left(x-2\right)^2+\left(y+2\right)^2=4$$
  

• 
  

  $$\left(x+2\right)^2+\left(y-2\right)^2=2$$
  

• 
  

  $$\left(x+2\right)^2+\left(y-2\right)^2=4$$
  

Ellipses

• We need to know 3 things to graph and ellipse

• The center (h,k)

• How long the major axis a is.

• How long the minor axis b is.

• Then the equation is written either as:

•  $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$  OR

•  $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$  

Focus

The focus for an ellipse is c away from the center, where we can find "c" by using the formula:

 $c^2=a^2-b^2$  

Multiple Choice

Which of the following equations represents the ellipse graphed to the left?

• $$\frac{\left(x-1\right)^2}{9}+\frac{\left(y-3\right)^2}{4}=1$$

• $$\frac{\left(x-1\right)^2}{4}+\frac{\left(y-3\right)^2}{9}=1$$

• $$\frac{\left(x+2\right)^2}{4}+\frac{\left(y+3\right)^2}{9}=1$$

Multiple Select

Which of the following equations represents an ellipse?

• $$\frac{\left(x+1\right)^2}{4}+\frac{\left(y-2\right)^2}{9}=1$$

• $$\frac{x^2}{25}+\frac{y^2}{9}=1$$

• $$\left(x+1\right)^2+4\left(y-2\right)^2=16$$

• $$\frac{\left(x-3\right)^2}{4}-\frac{\left(y+2\right)^2}{9}=1$$