Question

Show that the line x+y=1 touches the parabola y=x-x²

Answer 1

If line cuts the parabola then there will be two points
& if line touches the parabola then there will be only one
point.
Now,
Put y= 1-x in the equation of parabola
1-x = x- x^2
x^2 -2 x+ 1= 0
when solving quadratic
(x-1)^2 = 0
x = 1
Hence y= 0
As we see that there is only one solution
So this will be a tangent to parabola at (1, 0)

Simply
1 - x = x - x²
x² - 2x + 1 = 0
(x - 1)² = 0
x = 1

Line touches parabola at (1 , 0)