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

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The discussion focuses on finding the points where the tangent line to the equation x² + 4y + 22 = y² + 10x is horizontal. By differentiating the equation implicitly, the first derivative is established as (2x - 10) / (2y - 4). Setting the numerator to zero reveals that x = 5, leading to the points P(5, 1) and P(5, 3) where the tangent line is horizontal. The concept of tangents is clarified, emphasizing that a tangent line touches a curve at exactly one point, with its slope determined by the derivative at that point.

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Find the point(s) (x,Y) at which the tangent line to x^2+4y+22=y^2+10x is horizontal.
 
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What does it means when the tangent line is horizontal? what does the first derivative has to do with the "tangent"?
 
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.
 

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