Calculus Help about families of lines

[philly]

Could someone please help with this problem.

Find an equation for the family of lines tangent to the circle with center at the origin and radius 3??

I know that the equation of the circle is y^2+x^2=9 but where do I go from here.
Thanks
Philly

 

[tkhunny]

Could someone please help with this problem.

Find an equation for the family of lines tangent to the circle with center at the origin and radius 3??

I know that the equation of the circle is y^2+x^2=9 but where do I go from here.

You need to know the derivative at any point on the circle. Implicit differentiation should get you there for just about everything. Have you met implicit differentiation?

Note: There will be a little problem with the vertical tangents.

 

[philly]

Thanks for the response, but we are just starting Calculus I and have not come across def. yet. IS there a simplier way that can be used for a beginning calculus student??
Thanks
Philly

 

[tkhunny]

Well, then how about just a derivative. Have you had that?

Somehow, you need to find the slope of all those tangent lines. The derivative is the easiest way.

Maybe this is a problem designed to scare you just so you can be shown how easy it is with the derivative.

Geometry Hint: Tangent lines are perpendicular to Normal lines. Normal lines are through the Center of the circle and intersect the circle at the same point as the corresponding tangent line.

 

[pka]

There is a purely geometric way.
If (a,b) is appoint on the circle, then the line y=(b/a)x contains a radial segment.
Therefore a line tangent to the circle at (a,b) [b≠0] has slope (−a/b).
So the in general the tangent is y−b=(−a/b)(x−a).
Now recall that a<SUP>2</SUP>+b<SUP>2</SUP>=9.

 

[philly]

pka,
Thank you very much for your response.
We have not covered deviratives yet. We are only on Functions and graphing and I believe this is the way to go for now.
Looking forward to learning different approaches in the future.
Philly