Okay, thanks for your effort . Thoughts so far below:
For the normal to C at some point (at^2,2at) to pass through a point (h,k), h and k must satisfy. For exactly two normals to be drawn to the curve and pass through (h,k), there must be exactly two different values of t that satisfy this equation. For a cubic to have exactly two distinct roots, it must have a repeated root.
If we differentiate and set the derivative equal to 0,. This equation gives the value of t corresponding to the two turning points on the curve, one of which is our double root. Solving for t gives . In other words, when t is either of these values, the cubic we're interested in has exactly two real roots.
Thanks. Yes, you're right. I think the question assumes the real numbers though. I wasn't aware cubics had discriminants, either.
You haven't really answered my question though; I'm having trouble finding the Cartesian equation of the locus. When I did this question originally (a few months ago), I remember getting the result that the locus was another parabola (which I thought was rather nice). I can't seem to do it again this time round though
Hey, first post for a while.
I did this question a while ago, but I've lost my solution and I can't seem to reproduce it
Show that the equationhas exactly one real solution if . [this I can do: differentiating shows the curve is strictly increasing for non-negative p]
A parabola C is given parametrically bywhere a is a positive constant.
Find an equation which must be satisfied by t at points on C at which the normal passes through the point (h,k). Hence show that, if, exactly one normal to C will pass through (h,k) [this I can also do: the equation of the normal to C is , and we can just sub in (h,k) to find the equation]
Find, in Cartesian form, the equation of the locus of the points from which exactly two normals can be drawn to C.
I can't do the last bit this time round. I can see that the equation that must be satisfied by h and k needs to have a repeated root, but that's as far as I can get.
A lovely problem, taken from Advanced Problems in Mathematics, by Dr Siklos*
Two identical snowploughs plough the same stretch of road. The first starts at a timeseconds after it starts snowing, and the second starts from the same point seconds later, going in the same direction. Snow falls so that the depth of snow increases at a constant rate . The speed of each snowplough is , where is the depth (in metres) of the snow it is ploughing, and is a constant. Each snowplough clears all the snow. Show that the time at which the second snowplough has travelled a distance metres satisfies the equation: . Hence show that the snowploughs will collide when they have travelled metres.
*He adds in his discussion that this can be generalised to n identical snowploughs