In the xy-plane, the graphs of the linear equation y = 3x + k, where k is a constant, and … — SAT

In the $xy$ -plane, the graphs of the linear equation $y = 3x + k$ , where $k$ is a constant, and the quadratic equation $y = x^2 - 2x + 7$ intersect at two points. If the distance between the two points of intersection is $\sqrt{130}$ , what is the value of $k$ ?

A
1
B
-4
4Answer
D
10

Answer

The correct value of

k

4

. Equating the two equations yields the quadratic equation

x^2 - 5x + (7 - k) = 0

. Since the slope of the line is

3

, the difference in the y-coordinates of the intersection points is

3

times the difference in the x-coordinates. Applying the distance formula shows that the distance is equal to

\sqrt{10(x_2 - x_1)^2}

. Setting this equal to the given distance of

\sqrt{130}

yields

(x_2 - x_1)^2 = 13

. Substituting Vieta's relations (

x_1 + x_2 = 5

and

x_1 x_2 = 7 - k

) into the identity

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

results in

13 = 25 - 4(7 - k)

, which simplifies to

k = 4

Step-by-Step Solution

Equate the linear and quadratic equations to set up the intersection condition.

x^2 - 2x + 7 = 3x + k \dots \implies x^2 - 5x + (7 - k) = 0

The x-coordinates of the intersection points are the roots of this quadratic equation.

Express the distance between the two points

(x_1, y_1)

and

(x_2, y_2)

in terms of the difference of their x-coordinates.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(x_2 - x_1)^2 + (3(x_2 - x_1))^2} = \sqrt{10(x_2 - x_1)^2}

Since both points lie on the line

y = 3x + k

, the difference in their y-coordinates is

y_2 - y_1 = 3(x_2 - x_1)

Use the given distance to find the value of

(x_2 - x_1)^2

10(x_2 - x_1)^2 = 130 \implies (x_2 - x_1)^2 = 13

The distance between the intersection points is given as

\sqrt{130}

Apply Vieta's formulas and the algebraic identity for the squared difference of the roots to solve for

k

x_1 + x_2 = 5

and

x_1 x_2 = 7 - k

. Substituting these into

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

gives

13 = 5^2 - 4(7 - k) \implies 13 = 25 - 28 + 4k \implies 4k = 16 \implies k = 4

Vieta's formulas relate the coefficients of the quadratic equation to the sum and product of its roots.

Key Concept

Solving systems of linear and quadratic equations and finding constants based on geometric constraints.

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