Two Conditions, and the Point Must Be on the Curve

You'll learnhow to read an implicit slope as a fraction, top zero for a horizontal tangent and bottom zero for a vertical one, and why every candidate still has to be tested against the original equation.

A circle is not a function: one x can have two points on it, and each point has its own slope. The slope arrives as a fraction in x and y, so it has two ways of doing something interesting, and each way is only half an answer until the original equation has been asked whether the point is really on the curve. Eleven short steps; press Next for each one.

11 STEPS · 6 QUICK CHECKS · TOP ZERO IS HORIZONTAL · BOTTOM ZERO IS VERTICAL · THE POINT MUST BE ON THE CURVE · v2

x² + y² = 25 ⇒ dy/dx = −x/y · top zero ⇒ horizontal · bottom zero ⇒ vertical

Before you start

What you're looking at

The circle x² + y² = 25. Its slope is dy/dx = −x/y, a fraction that needs a whole point before it gives a number.

The question

Where is the tangent horizontal, where is it vertical, and how do you know a candidate point is really on the curve?

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