The equation of the tangent line to the parabola y=x+1 passing through the point (−8,0) is....
Explanation
The tangent line has the general equation y=mx+c which passes through the point (−8,0). Substitute the point to get
y=mx+c
0=−8m+c
c=8m
Thus the tangent line becomes
y=mx+8m
x=my−8m
Change the parabola function into x form
y=x+1
x=y−1
x=(y−1)2
Substitute the line into the parabola
my−8m=(y−1)2
my−8m=y2−2y+1
y−8m=my2−2my+m
0=my2−2my−y+m+8m
my2−(2m+1)y+9m=0
This becomes a quadratic equation with a=m, b=−(2m+1), c=9m. The condition for tangency is D=0
b2−4ac=0
{−(2m+1)}2−4m(9m)=0
4m2+4m+1−36m2=0
−32m2+4m+1=0
32m2−4m−1=0
(8m+1)(4m−1)=0
m=−81∪m=41
Because the form of y=x+1 is always an increasing function, then m=41 satisfies. So the equation of the tangent line is
y=mx+8m
y=41x+8⋅41
y=41x+2
4y=x+8
4y−x−8=0