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:

adore.nuk
Jun 8, 2017

#1**+1 **

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

CPhill
Jun 8, 2017