# Thread: Coordinates where tangent line is vertical?

1. ## Coordinates where tangent line is vertical?

Hey anyone know how I can solve this?

The equation is (x^2)+2x+(y^4)+4y=5

How do I find the coordinates of the two points on the curve where the line tangent to the curve is vertical?

Also, is it possible for this curve to have a horizontal tangent line at the points where it intersects the x-axis? Why or why not?

Any explanations would be great so I could learn how to approach this problem in the future. Thanks!

2. Originally Posted by painterchica16
How do I find the coordinates of the two points on the curve where the line tangent to the curve is vertical?
$x^2 + 2x + y^4 + 4y = 5$

These would be the points at which x' = dx/dy = 0. Differentiating implicitly with respect to y, we have

$2xx' + 2x' + 4y^3 + 4 = 0$

$(2x + 2)x' = -4y^3 - 4$

$x' = \frac{-4y^3 - 4}{2x + 2}$

Setting $x' = 0$ yields

$\frac{-4y^3 - 4}{2x + 2} = 0 \implies -4y^3 - 4 = 0$

so $y^3 = -1$; y = -1.

Substituting back into the original equation,

$x^2 + 2x + (-1)^4 + 4(-1) = 5$

$x^2 + 2x + 1 - 4 = 5$

$x^2 + 2x - 8 = 0$

$(x + 4)(x - 2) = 0$

$x = -4, 2$

So the two points are (-4, -1), and (2, -1).