Find the coordinates of the intersections.

Question

Find the coordinates of the intersections.

The equation of a circle of radius r with its center at the point (h,k) is given by

(x-h)^2+(y-k)^2=r^2

Plot the following functions in a grid, and algebraically find the coordinates of their intersections.
Leave answer in exact form.

(x+2)^2+(y-1)^2=3^2

y=x^2+4x+7

Answer 1

( x + 2 )² + ( y - 1 )² = 3²  ;  y = x² + 4x + 7

 

You have a circle and a parabola;  you are looking for their intersection;

they can intersect in the following # of points: 0 , 1, 2, 3, or 4

 

You need to solve the two equations simultaneously

You could substitute for one of the variables in one of the equn.

 

Looking for an easy way, using:

 

y = x² + 4x +7

 

y - 7 = x² + 4x

 

Complete the square making sure you do not change the value 

of the right side of equation

 

y - 7 = ( x + 2 )² - 4

 

y - 3 = ( x +2 )²

 

Now you can substitute this value for ( x + 2 )²  into the equation for 

the circle

 

       y - 3 + ( y - 1 )² = 3²

 

     y -3 + y² -2y + 1 = 9

 

                 y² -y -11 = 0

 

                      y² - y = 11

 

                 ( y - .5)² = 11 + .25

 

                      y - .5 = ± √11.25

 

                            y = .5 ± √ 11.25

 

Substitute this value into either equation a solve for x; you should end

up with either 2 or 4 different points

 

When they ask for answers to be in exact form, they want them to stay

with the "√ " sign