# Math Help - Largest Value of m?

1. ## Largest Value of m?

Hi, I was told that this was a simple calculus problem so I thought I should post it here.

What is the largest value of m such that the graph of the equation y=mx meets the graph of the equation (x-10)^2 + (y-5)^2 =4

Thanks for any help.

2. $(x-10)^2+(y-5)^2=4$ and $y=mx$

$(x-10)^2+(mx-5)^2=4$

$x^2-20x+100+m^2x^2-10mx+25=4$

$x^2(1+m^2)-x(20+10m)+121=0$

Then use the quadratic equation to find the largest value of m.

3. Ok, I did the quadratic equation for it, but now am stumped. I have:

(-444x-464mx)/(4x^2 +4m^2 x^2) and (-484x-484mx)/(4x^2 +4m^2 x^2)

What do I do now?

4. Originally Posted by Showcase_22
$(x-10)^2+(y-5)^2=4$ and $y=mx$

$(x-10)^2+(mx-5)^2=4$

$x^2-20x+100+m^2x^2-10mx+25=4$

$x^2(1+m^2)-x(20+10m)+121=0$

Then use the quadratic equation to find the largest value of m.
Just to add:

Draw a picture to see why you want the line to be a tangent to the circle. Therefore you want the quadratic equation to have only one solution. Therefore you require the value of the discriminant to be equal to zero.

Note that the discriminant is $(20 + 10m)^2 - 4(1 + m^2)(121)$.

5. (-444x-464mx)/(4x^2 +4m^2 x^2) and (-484x-484mx)/(4x^2 +4m^2 x^2)
This isn't right. Applying the quadratic formula should give an expression only in terms of m (ie. no x's).

See if you can get the discriminant the same as Mr Fantastic and follow his instructions from there.