INFINITY GRADIENT ? A level edexcel maths help please C4

https://cdn.savemyexams.co.uk/wp-content/uploads/2018/11/A9.8-Implicit-Topic-Booklet-1-A-Level-Edexcel-Maths-Pure.pdf

question 1 b
how are you meant to find the farthest point on a curve ? i cant find an example in my textbook to explain it to me
in the mark scheme it says to equal the derivative to infinity but why ?

Link does not work.

can you see it now ?

It tells you the points furthest west and east respectively are P and Q. These have tangents at that point represented by the dotted line. If you cast your mind back to first doing gradients, a horizontal line has a gradient of 0 and the steeper it gets, the greater the gradient gets. However if you try calculate the gradient of a vertical line by conventional methods, you will find the for the y coordinate to increase the change in x is 0. So dy/dx has to be infinity as a straight vertical line represents an infinite gradient. If anymore help is needed or clarification on what I've said then feel free to ask, my explanation is a bit strange i admit.

Gradient is zero, so the numerator is zero.

why is it zero ?

The grad is infinite at these two points, so the denominator must be zero.
This gives a single equation in X and y.
Sub this into the equation of the ellipse to determine the value(s) of the variable(s).
Post your working if there is a problem.

So what you're basically saying is the 1/0 = infinity, right?
Anytime I mention that idea to my professor he thinks I've lost a screw.

Strictly speaking, divide by zero is undefined. In this case as the denominator approaches zero, the gradient increases without bound. So it approaches infinite.
Ask your prof what he/she means, if you domdo understand what they're saying?

3 years later, another poor student (me) is again plagued by this problem

Generally best to not bump old threads, but what are you plagued by?

this question turned up in a homework y=x for part b doesn't seem to make sense

y=x is indeed incorrect for part b. What are you trying to do and what will make it correct?