what is the value of k?

Question

Fink k if the curve y = x^2 + k is tangent to the line y = 2x



I know the answer, but please EXPLAIN why the answer is what it is.

Answer 1

An intersection point is a point where the line touches the curve.
If the line cuts across the curve, there will be two intersection points.

However, if the line just touches the curve once, then we say that it is tangent.
There is only one intersection point.

To find intersection point(s), you solve the system
y = x^2 + k
y = 2 x

Any point that satisfies the system MUST be on the line AND on the curve (therefore, an intersection point).

Because you have 3 unknowns (x, y, k) and only two equations, you will find a family of solutions. In that family, you will control the value of k.

Your task is to find the value of k that will give you the 'unique' solution where there is only one intersection point.

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substituting y = 2x

2x = x^2 + k
x^2 - 2x + k = 0

quadratic

x = [ 2 +/- SQRT(4 - 4k) ]/2

+/- means plus or minus.

Anytime the square root (SQRT) has a strictly positive value, you will have 2 answers (therefore 2 intersection points : the line cuts the curve).

Anytime the value in the square root is negative, then the root does not exist and you will have zero intersection (the line misses the curve)

If you set k so that the value inside the square root is exactly 0, then your answer will include a +/- 0 which means that 'both' answers are the same: there is only one intersection point. The line is tangent.

Here, if you set k=1, then 4 - 4k becomes 0 and the solution becomes:

x = [ 2 +/- SQRT(0) ]/2
x = [ 2 +/- 0 ]/2
x = [ 2 ] /2 = 1
y = 2x = 2
the point where the line is tangent to the curve is (1, 2).
The bottom of the curve y = x^2 + k is at (0, 1)

Answer 2

K=I SINCE IF THE LINE IS A TANGENT THEN IT ONLY MEETS THE CURVE AT ONE POINT ie the discriminant of the equation
x squared -2x +k=0 must be zero for repeated roots

Answer 3

the line is tangent to a parabola iff they have ONLY 1 common point.
That means that the equation
2x = x^2 + k
mast have only 1 solution
it's discriminant D = 4-4k have to be 0
so k = 1.

Answer 4

Put y=2x in y=x^2+k
2x=x^2+k
x^2-2x+k=0. this is a Quadratic Eq.
Itsdiscriminant=(-2)^2-4k
for y=2x to be atangent to y=x^2+k,
the two roots of this quadratic Eq. should be equal, so its
discriminant=0
4-4k=0
k=1. ANS