Hi **xwrathbringerx**,

Below, you will find solutions to *Problems 1* and *2*.

Originally Posted by

**xwrathbringerx** 1.The tangent at P on the ellipse x^2/a^2 + y^2/b^2 =1 cuts the x axis at T and the perpendicular PN is drawn to the x-axis. If O is the origin, prove that ON * OT = a^2.

Consider the ellipse

Let , then the equation of the tangent line at the point is given by

or eqivalently

......................(1)

Actually, obtaining the equation of the tangent line is not very simple (just makes use of a simple idea but too much computations) but If you need I can also show it.

To find , we have to plug into (1), i.e., put and .

Therefore, we get

.............................(2)

On the other hand, if is the projection of , we must have

.........................................(3)

And finally, from (2) and (3), we can find that

As a note, the value is called the *subtangent length* of the point (see *Figure 1*).

Originally Posted by

**xwrathbringerx** 2. Find the equation of the normal l to 9x^2 + 25y^2 = 225 at P(3, 2 2/5). This normal cuts the x-axis at G and N is the foot of the perpendicular drawn from P to the x-axis. Find GN.

You can see from the solution of *Problem 1* that the slope of the tangent line at is

Hence, and the point the normal line must have the slope

since .

Then, the equation of the normal line is at the point is

or equivalently

......................(4)

where

To find , use (4) with and , then we get

...........................(5)

And the foot must satisfy

.........................................(6)

So that (5) and (6) yield

......_

......_

This value is called the *subnormal length* of the point (see *Figure 1*).

You may put above , and to get your answer.

**Figure 1**: Subtangent, and subnormal lengths.

Think of the remaining ones with drawing figures,

if you wont be able to succeed I can give a hand again.