Help Maths Revsion-Coordinate Geometry of a Circle

[moogoomonkey]

Homework Statement

Ok hey guys this is my first post so be nice ;).
I just wanted to know how to see if a line only intersects the circle once (ie a tangent)
I know about double intersections with the quadratic and no intersection with a negative surd in the quadratic equation but is there a method (apart from sketches ie algebra) to find out if it only intersects once?

Homework Equations

Circle C has the equation x^2+y^2=18
A line L has the equation y=x+6
a) Prove that the line intersects the circle in exactly one place
b) What name is given to such a line

The Attempt at a Solution

x^2+y^2=18

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

2x^2+12x+36=18

x^2+6x+18=18

x^2+6x=0...

What now??

Thanks in advance guys! =D

 

[Mentallic]

Homework Statement

Ok hey guys this is my first post so be nice ;).
I just wanted to know how to see if a line only intersects the circle once (ie a tangent)
I know about double intersections with the quadratic and no intersection with a negative surd in the quadratic equation but is there a method (apart from sketches ie algebra) to find out if it only intersects once?

Algebra and sketching are totally different things. Sketching is only as accurate as you can draw (and it'll never be perfect) while algebra is perfect.

Homework Equations

Circle C has the equation x^2+y^2=18
A line L has the equation y=x+6
a) Prove that the line intersects the circle in exactly one place
b) What name is given to such a line

The Attempt at a Solution

x^2+y^2=18

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

2x^2+12x+36=18

x^2+6x+18=18

x^2+6x=0...

What now??

Thanks in advance guys! =D

You made a mistake on the red line. You divided the left side through by 2, but not the right.
But either way, you're pretty much there, you just need to realize what the algebra is telling you. Solving that quadratic (which is wrong) by factorization gives us x(x+6)=0, so x=0,-6. This would tell us that the line and the circle intersect at x=0 and at x=-6, and plugging those values back into either the equation of the line or circle will give us the corresponding y-values. From this you would deduce that the line is not tangent to the circle since it cuts it in 2 places.
BUT since you made an error in your calculations, this answer is wrong (but the process of finishing the problem is still valid).

 

[moogoomonkey]

Thank you so much =D

 

[moogoomonkey]

Ok once i sat down and had a proper look armed with new knowledge of my mistake (thanks to Mentallic) I solved the problem.


x^2+y^2=18

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

2x^2+12x+36=18

x^2+6x+9=0

this thus factorises as

(x+3)(x+3) and hence it only intersects once as the x values are the same (x=+3)since the factorisation is a perfect square

Thanks anyway Mentallic =D

 

[moogoomonkey]

Homework Statement

I just wanted to know how to see if a line only intersects the circle once (ie a tangent)

Ooops i guess i answered my own question since the name given to a line with 1 intersection is a TANGEN! Stupid me =S

 

[Mentallic]

(x+3)(x+3) and hence it only intersects once as the x values are the same (x=+3)since the factorisation is a perfect square

Just as a heads up, it's customary to express that as (x+3)² (now it's looking more like a perfect square ). Also, the answer is x=-3 and not 3 because we now have (x+3)²=0 thus x=-3 satisfies this equation.

Ooops i guess i answered my own question since the name given to a line with 1 intersection is a TANGEN! Stupid me =S

Yep