Write the equation of the line tangent to at the point where the given curve intersects the x-axis.

Yay, calc

we need to take take the implicit derivitive
to evaluate the slope, we need the point

when the line crosses the x axis, the value of y is 0
find the x value when y=0 to find the point, (x,0)

0+0+4x=8
4x=8
x=2
(2,0) Is the point

so

the derivitive is

minus 2xy both sides

minus 4 both sides

undistribute dy/dx

divide both sides by x²+3y²

now find the slope at the point (2,0)

sub 2 for x and 0 for y




slope is -1

use point slope form
the equation of a line that has a slope of m and passes through the point (x1,y1) is y-y1=m(x-x1)
slope is -1 and point is (2,0)

y-0=-1(x-2)
y=-1(x-2)
or
y=-x+2