Find the coordinates of intersection

Question

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

 

Find coordinates of intersections of the following functions

 

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

 

y=x^2+4x+7

Answer 1

If you look at it graphically, the you have a parabola intersecting a circle.  When this happens, you will at least 4 points of intersections.

 

 

First expand the circle equation so that we can obtain something to substitute into the other equation.

 

x² + 4x + 4 + y² - 2y + 1 = 9

 

x² + 4x + y² - 2y + 5 = 9

 

x² + 4x + y² - 2y = 4

 

x² + 4x = -y² + 2y + 4

 

 

(x² + 4x) is a term that is the parabola equation.

 

 

Next, we substitute this equation into the parabola equation.

 

y = -y² + 2y + 4 + 7

 

y = -y² + 2y + 11

 

 

Move all terms to the left side of equation.

 

y² - y - 11 = 0

 

 

Now we can solve for y from this equation.  Use the quadratic formula:

 

y = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 1

b = -1

c = -11

 

Plug in these values into the formula.  You will get two y-values.  Then substitute those y values into the circle equation to solve for x.