Find all points on the graph of the equation x2 - y2= 2x + 4y where the tangent line is horizontal. Does the graph have any vertical asymptote? Sketch the graph.
Answers
Given : x² - y² = 2x + 4y
To find : all points on the graph of the equation where the tangent line is horizontal.
Solution:
x² - y² = 2x + 4y
2x - 2y(dy/dx) = 2 + 4(dy/dx)
=> 2x - 2 = (dy/dx)(2y + 4)
=> x - 1 = (dy/dx)(y + 2)
=> (dy/dx) = (x - 1)/(y + 2)
tangent line is horizontal
=> Slope = 0
(dy/dx) = 0 at tangent point
=> (x - 1)/(y + 2) = 0
=> x = 1
x² - y² = 2x + 4y
=> 1 - y² = 2 + 4y
=> y² + 4y + 1 = 0
=> y = (- 4 ± √12)/2
=> y = - 2 ± √3
{Points on the graph of the equation where the tangent line is horizontal.
(1 , -2 +√3) & (1 , -2 -√3)
y = ±√(x²- 2 x + 4) - 2 hence no vertical asymptote
Learn more:
If y = tan-1 x, show that ( 1 + x2) d2y / dx2 + 2x dy/dx = 0
https://brainly.in/question/3206880
The tangent to the curve y = x2 + 3x will pass through the point 0, - 9 ...
https://brainly.in/question/11223243
