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1. The equation of a circle is x^2 + y^2 - 4x + 2y - 11 = 0. What are the center and the radius of the circle? Show your work.

Answer:

 

2. Write the equation of the circle in general form. Show your work.

3.

Write the equation of a parabola with focus (-2,4) and directrix y = 2. Show your work, including a sketch.

Answer:

 Jun 8, 2017
 #1
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1)   x^2 + y^2 - 4x + 2y - 11 = 0     add  11  to both sides  and  rearrange 

 

x^2 -  4x  + y^2  +  2y    =  11

Complete the square on x......take (1/2)  of 4  = 2, square it  = 4  and add to both sides

 

x^2 - 4x + 4  + y^2 + 2y  =  11 + 4

Complete the square on y.....take (1/2) of 2 = 1, square it =  1  and add to both sides

 

(x^2 - 4x + 4)  +  ( y^2 + 2y + 1)  =  11 + 4 + 1

 

Factor the perfect square trinomials in each set of parentheses and simplify the right side

 

(x - 2)^2  +  (y + 1)^2   =  16

 

The center  is   ( 2, -1)   and the radius  =   4

 

 

2) The center  is   (-1,1)   and the radius  is 3

So...the equation is   

(x + 1)^2  +  ( y - 1)^2  =  9

 

 

3)  We have the form

 

4p(y - k)  =  (x - h)^2    where (h, k)  is the vertex  and p is the distance between the vertex and the focus

 

The  vertex  can be found  as

 

( -2,   [ y coordinate of the focus + y value of the directrix]/ 2 )  =

 

(  - 2, [ 4 +2] / 2 )    =   ( -2, 6/2)  =   ( -2, 3)

 

So....the distance between the focus (-2, 4) and the vertex (-2, 3)   =  1  = p

 

So......the equation  becomes

 

4(1) ( y -3)  = ( x + 2)^2

 

4(y - 3)  =  ( x + 2)^2

 

Here's a graph  : https://www.desmos.com/calculator/tyvpqyor1i

 

 

cool cool cool

 Jun 8, 2017

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