Question 1200188: Determine the set of values of the constant k for which the line y = 4x + k does not
intersect the curve y = x^2 Found 2 solutions by Alan3354, ikleyn:Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website! Determine the set of values of the constant k for which the line y = 4x + k does not
intersect the curve y = x^2
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The slope, m, of y = 4x + k is 4.
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Find the point on y = x^2 where the slope is 4:
y' = 2x is the slope at any point on y = x^2
2x= 4 at x = 2
The point (2,4) is where the line is tangent to the parabola.
---> k < -4
You can put this solution on YOUR website! .
Determine the set of values of the constant k for which the line y = 4x + k does not
intersect the curve y = x^2
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Here an Algebra solution is placed.
If the/an intersection point does exist, then an equation
x^2 = 4x + k (1)
has a real solution. This equation is equivalent to
x^2 - 4x - k = 0.
The discriminant of this quadratic equation is
d = b^2 - 4ac = (-4)^2 - 4*1*(-k) = 16 + 4k.
The discriminant is positive if and only if
16 + 4k >= 0
4k >= -16
k >= -16/4 = -4.
Thus a real solution to equation (1) does exists if and only if k >= -4.
If k < -4, there is no real solution for equation (1), so the intersection does not exist.
ANSWER. The set of values of "k" is { k < -4 }.
