A line is defined by the equation y = 3x + k, where k is a constant. This line intersects … — ACT

A line is defined by the equation $y = 3x + k$ , where $k$ is a constant. This line intersects the parabola $y = x^2 - x + 2$ at two distinct points, $P$ and $Q$ . If the midpoint of the line segment $PQ$ lies on the line $y = 2x + 7$ , what is the value of $k$ ?

Answer: 5

Answer

The value of

k

5

Equating the equations of the line and the parabola gives a quadratic equation

x^2 - 4x + (2 - k) = 0

. The average of the roots of this quadratic equation gives the

x

-coordinate of the midpoint,

x_m = 2

. Substituting this into the first line's equation gives the

y

-coordinate of the midpoint,

y_m = 6 + k

. Since the midpoint

(2, 6+k)

lies on the line

y = 2x + 7

, we substitute these coordinates to get

6 + k = 11

, which simplifies to

k = 5

Step-by-Step Solution

Equate the equations of the line and the parabola.

x^2 - 4x + (2 - k) = 0

To find the

x

-coordinates of the intersection points

P

and

Q

Determine the sum of the

x

-coordinates and find the midpoint's

x

-coordinate.

x_m = 2

By Vieta's formulas, the sum of the roots of the quadratic equation is

4

. The

x

-coordinate of the midpoint is the average of these roots:

4 / 2 = 2

Find the

y

-coordinate of the midpoint.

y_m = 6 + k

Because the midpoint lies on the line segment

PQ

, its coordinates must satisfy the equation of the line passing through

P

and

Q

, which is

y = 3x + k

Substitute the midpoint coordinates into the second line's equation and solve for

k

k = 5

We are given that the midpoint lies on the line

y = 2x + 7

Key Concept

Systems of Linear and Quadratic Equations and Midpoint Properties

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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