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:
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