Integration help please (2 Viewers)

[peterkim95]

Ive got an exam tomorrow, but i dunt know how you do this question :mad1:
prove that the line y = x + 2 is a tangent to the parabola y= x^2 - 5x + 11
Thanks for your help!

 

[watatank]

solve them simultaneously...well at least combine the two equations and make it one...

y = x + 2
y= x^2 - 5x + 11

x + 2 = x^2 - 5x + 11
x^2-6x-9 = 0

discriminant of x^2-6x+9

= b^2-4ac
=36-4(1)(9)
=0

since the discriminant = 0, y = x+2 only intersects the parabola at ONE point. if the discriminant were negative it wouldn't touch the parabola at all and it were positive it would touch at two points.

therefore it must be a tangent.

hope that helps :wave:

 

[meomeo]

internet maths it too hard D':

 

[peterkim95]

thank you soo much1

 

[ThuanSUX]

An alternative solution to what Watatank said would be:

1. Combine the equations
2. Factorize

y = x+2
y = x^2-5x+11

x^2-5x+11 = x+2
x^2-6x+9 = 0
(x-3)(x-3) = 0
(x-3)^2 = 0
thus, x=3 ONLY

Resubstituting it into the first equation to obtain a y-coordinate and you get the point (3,5). Hence, the two equations only intersect at one point, making y=x+2 a tangent of y=x^2-5x+11.