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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.

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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.

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.

Jun 8, 2017

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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

Jun 8, 2017