Finding Horizontal Tangents of x^2+4y+22=y^2+10x

[candynrg]

Find the point(s) (x,Y) at which the tangent line to x^2+4y+22=y^2+10x is horizontal.

 

[Pyrrhus]

What does it means when the tangent line is horizontal? what does the first derivative has to do with the "tangent"?

 

[wisredz]

Differentiating this equation implicitly with respect to x yields 2x-10/2y-4. Thus we see that 2x-10=0 (2y-4 can't equal zero, that would be undefined) From here x=5. So the point is P(5,y). set x=5 in the original equation, solve for y and there you got the y. This turns out that that y has two values for x=5, and those being 1 and 3.

Btw, tangent comes from the latin word tangens, which means touch. So a tangent line touches a curve at just one point. And the derivative of the curve at that point gives us the slope of the tangent line.