Hi everyone. I can't fully describe things in the title, so here is the full question:

is any point on the rectangular hyperbola . The tangent at P cuts the y-axis at T and the normal at P cuts the x-axis at N. Find the Cartesian equation of the locus of the midpoint of TN.

My attempted solution: First, I define T to be the point and N to be the point .

To find the Gradient of the tangent, I implicitly differentiate the hyperbola. Therefore:

Here, I substitute the values of P into the gradient, and try to solve for PT and PN.

So using this to solve for PT: The gradient of the tangent using the values of P would be This makes the equation PT to be:

Therefore:

The gradient for the normal would be , and the equation would be

Using the found values of p and q, let M be the midpoint of TN. Therefore,

Here I look like this.

How do I find the gradient to the midpoint so that I can find the Cartesian equation. After being like for some time, I turned over to the answer and it was given as:

What am I doing wrong? Or can c be substituted into the equation later? Can I have some pointers please?

Thanks in advance!!