A line is defined by the equation y = 2x + k, where k is a constant. A parabola is defined… — ACT

A line is defined by the equation $y = 2x + k$ , where $k$ is a constant. A parabola is defined by the equation $y = x^2 - 4x + 14$ . If the line and the parabola intersect at exactly one point in the standard $(x, y)$ coordinate plane, what is the value of $k$ ?

Answer: 5

Answer

The value of the constant

k

must be 5.

To find the intersection of the line and the parabola, set their equations equal to each other:

x^2 - 4x + 14 = 2x + k

. Rearranging this into standard quadratic form gives

x^2 - 6x + (14 - k) = 0

. For the system to have exactly one solution, the discriminant of this quadratic equation must be zero. The discriminant is

b^2 - 4ac = (-6)^2 - 4(1)(14 - k) = 36 - 56 + 4k = 4k - 20

. Setting

4k - 20 = 0

yields

k = 5

Step-by-Step Solution

Equate the linear and quadratic expressions to find their intersection.

x^2 - 4x + 14 = 2x + k

Setting the two equations equal to each other allows us to find the

x

-coordinates of any intersection points.

Rearrange the terms to write the equation in standard quadratic form,

ax^2 + bx + c = 0

x^2 - 6x + (14 - k) = 0

Subtracting

2x

and

k

from both sides simplifies the equation into a quadratic form where

a = 1

b = -6

, and

c = 14 - k

Apply the condition for exactly one intersection point by setting the discriminant to zero.

(-6)^2 - 4(1)(14 - k) = 0

A quadratic equation has exactly one real root if and only if its discriminant,

b^2 - 4ac

, is equal to zero.

Solve the linear equation for

k

k = 5

Expanding the equation yields

36 - 56 + 4k = 0

, which simplifies to

4k - 20 = 0

, leading to

k = 5

Key Concept

Determining the condition for a linear equation to be tangent to a quadratic equation by setting the discriminant of their intersection equation to zero.

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