How to show algebraically that Q reaches its maximum as P approaches O

[Manni]

I need help with this question. I understand that logic behind it; as P approaches O the value the right bisector Q reaches it's maximum. I don't know how to show this algebraically however. Help?

 

[micromass]

We give P the coordinates (x,y). We know that P is on the parabola, so we know that

$$P=(x,x^2)$$

Now, can you find the coordinates of the point Q??

 

[Manni]

Ok, so the bisector would intersect P at point x/2. Because I know that the x coordinate on Q is 0, how would I find it's y?

 

[micromass]

Answer the following steps:

- Find the midpoint M between 0 en P.
- Find the equation of the line L going through 0 and P
- Find a vector perpendicular to the line L
- Construct the equation of the line R going through M and perpendicular through L
- Find Q as the intersection between R and the y-axis.

All of these questions involve nothing more than 10th grade geometry. So you should be able to complete these easily.

 

[Manni]

How do I show algebraically that Q approaches infinity as P approaches the origin?

 

[micromass]

How do I show algebraically that Q approaches infinity as P approaches the origin?

It doesn't approach infinity.

Did you find the expression for Q?