# At what point(s) does the line #x + y =6# and the circle #x^2 + y^2 = 18# intersect?

##### 2 Answers

The line will be tangent to the curve it it intersects the curve at a repeated solution.

Solving simultaneously:

# \ \ \ x+y=6 #

# x^2+y^2 = 18 #

Thus by eliminating

# x^2 + (6-x)^2 = 18 #

# :. x^2 + 36 -12x + x^2 = 18 #

# :. 2x^2 -12x + 18 = 0 #

# :. x^2 -6x + 9 = 0 #

# (x-3)^2 = 0 => x=3 # , a repeated root

And with

# y = 6-- 3 = 3#

So the line touches the curve at

graph{ (x^2+y^2 - 18)(x+y-6)=0 [-10, 10, -5, 5]}

**Graphically**

We can graph the circle

We can see that the line touches the circle at one point, namely

**Algebraically**

We solve the following system of equations to check if the two curves indeed intersect.

#{(x^2 + y^2 = 18), (x + y = 6):}#

Solving:

#y = 6 - x#

Substituting:

#x^2 + (6 - x)^2 = 18#

#x^2 + 36 - 12x + x^2 = 18#

#2x^2 - 12x + 18 = 0#

#x^2 - 6x + 9 = 0#

#(x -3)(x - 3) = 0#

#x = 3#

Now solving for

Hopefully this helps!