Curves + Tangents (1 Viewer)

[xwrathbringerx]

Hi guys

I've been doing some revision for my yearlies and I'm stuck on these questions:

1) Find the values of m for which the curves y = mx and y = abs(sinx) have only one common point.

2) Show that, if the line y = mx + c is a tangent to the curve 4x^2 + 3y^2 = 12, then c^2 = 3m^2 + 4.

3) The curves with equations y = 4/x + 2 and y = ax^2 + bx + c have the following properties:
1. there is a common point where x = 2
2. there is a common tangent where x =2
3. both curves pass through the point (1,6)
Find the values of a, b and c.

Could you please help me out?

Thanx a lot.<!-- google_ad_section_end -->

 

[gurmies]

 

[Michaelmoo]

3) When x = 2, y = 4/2 + 2 = 4
so:

4 = 4a + 2b + c ....................(1)

For the quadratic, take dy/dx you get dy/dx = 2ax + b
Now when x=2, dy/dx = 4a + b

Now for the hyperbole, take dy/dx you get dy/dx = -4/(x^2)
Now when x=2, dy/dx =-1

Because its a common tangent at that point, gradient of tangent at that point are equal. i.e.
4a + b = -1 ......................... (2)

For the last part, sub that into the quadratic, you get:
a + b + c = 6 .....................(3)

So you solve (1), (2) and (3) simultaneously, you should get a = 1, b = -5, c = 10

 

[Michaelmoo]

1) Basically, if you draw both graphs, you see that the line can only pass through the origin.

Now, this gradient should be greater than that of the curve sinx, if you take the derivative of sinx, you get dy/dx = cosx, now dy/dx is a maximum when x = 0 and dy/dx = 1

So this gradient must be greater than 1. BUT when the gradient is one, there is only one point of intersection anyway. So the gradient is grater than or equal to one.

If you keep rotating the line anticlockwise, eventually your gradient will be a really large negative number and it approaches -1 (During this rotation there is still only one point of intersection). This allows you to deduce that the gradient could also be less than (or equal to) negative one.

In summary m>1 or m<-1

Sorry my explanation is dodgy I know. Maybe someone can give you a better explanation or prove it mathematically...