In the standard (x, y) coordinate plane, a line L passes through the points (a, a^2) and (… — ACT

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the points $(a, a^2)$ and $(b, b^2)$ , where $a$ and $b$ are distinct real numbers. The slope of line $L$ is $8$ . If the midpoint of the line segment connecting these two points lies on the line $y = 5x - 1$ , what is the $y$ -coordinate of this midpoint?

Answer: 19

Answer

The

y

-coordinate of the midpoint is

19

Applying the slope formula to the points

(a, a^2)

and

(b, b^2)

yields \frac{b^2-a^2}{b-a} = a+b = 8. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4. Substituting this value into the equation

y = 5x - 1

gives

y = 5(4) - 1 = 19

Step-by-Step Solution

Express the slope of line

L

in terms of

a

and

b

and set it equal to the given slope.

a + b = 8

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

. For the points

(a, a^2)

and

(b, b^2)

, the slope is \frac{b^2 - a^2}{b - a}. Factoring the numerator gives \frac{(b-a)(b+a)}{b-a} = a + b. Since the slope is given as

8

, we establish

a + b = 8

Find the

x

-coordinate of the midpoint of the segment connecting the two points.

x_M = 4

The midpoint formula for the

x

-coordinate is

x_M = \frac{x_1 + x_2}{2}

. For our points,

x_M = \frac{a+b}{2}

. Substituting the value

a+b = 8

gives

x_M = \frac{8}{2} = 4

Determine the

y

-coordinate of the midpoint using the line equation.

y_M = 19

The midpoint lies on the line

y = 5x - 1

. Substituting

x_M = 4

into this equation gives

y_M = 5(4) - 1 = 19

Key Concept

Slope of a Line

Alternative Method

Let

a

and

b

be the real roots of the quadratic equation

t^2 - 8t + 13 = 0

. By Vieta's formulas,

a+b = 8

and

ab = 13

. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4, and the

y

-coordinate is \frac{a^2+b^2}{2} = \frac{(a+b)^2 - 2ab}{2} = \frac{64 - 26}{2} = 19. Since the point

(4, 19)

satisfies

y = 5x - 1

, this confirms the existence of valid real coordinates

(a, a^2)

and

(b, b^2)

that produce the midpoint on the line.

Estimated Time:2m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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