In the standard (x, y) coordinate plane, the point M is the midpoint of the line segment w… — ACT

In the standard $(x, y)$ coordinate plane, the point $M$ is the midpoint of the line segment with endpoints $P(-4, -2)$ and $Q(4, 2)$ . A second line segment is drawn from $M$ to a point $R(x, y)$ such that the length of the segment $MR$ is $8$ . If the midpoint of the segment $MR$ lies on the line $3x - 4y + 12 = 0$ , what is the smallest possible value of $x$ ?

Answer: -8

Answer

The smallest possible value of

x

-8

By finding the midpoint

M(0,0)

PQ

and writing the midpoint of

MR

\left(\frac{x}{2}, \frac{y}{2}\right)

, we substitute this into the line equation to find

y = \frac{3}{4}x + 6

. We then substitute this into the distance formula equation

x^2 + y^2 = 64

to get the quadratic equation

25x^2 + 144x - 448 = 0

, which yields the solutions

x = -8

and

x = 2.24

. The smallest possible value is

-8

Step-by-Step Solution

Calculate the coordinates of the midpoint

M

of segment

PQ

M = (0, 0)

The midpoint formula states that the midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Express the midpoint

N

of segment

MR

in terms of

R(x, y)

N = \left(\frac{x}{2}, \frac{y}{2}\right)

The midpoint of

M(0, 0)

and

R(x, y)

is found by averaging their coordinates.

Substitute the coordinates of

N

into the equation of the line

3x - 4y + 12 = 0

y = \frac{3}{4}x + 6

Since the midpoint

N

lies on the line, its coordinates must satisfy the line's equation, which gives a linear relationship between

x

and

y

Set up the equation for the distance

MR = 8

x^2 + y^2 = 64

The distance formula between

M(0, 0)

and

R(x, y)

d = \sqrt{x^2 + y^2}

, and squaring both sides gives

x^2 + y^2 = d^2

Substitute

y = \frac{3}{4}x + 6

into the distance equation and solve the quadratic equation.

x = -8

and

x = 2.24

Substituting the linear relationship into the quadratic circle equation gives a single quadratic equation in terms of

x

, which can be solved using the quadratic formula.

Determine the smallest value of

x

from the two possible solutions.

-8

Comparing the two solutions,

-8

is smaller than

2.24

Key Concept

Distance and Midpoint Formulas

Alternative Method

Instead of solving algebraically, one can scale the line

3x - 4y + 12 = 0

by a factor of 2 centered at the origin

M(0,0)

to directly obtain the line equation on which

R

lies:

3x - 4y + 24 = 0

. Then, find the intersection of this line with the circle

x^2 + y^2 = 64

using substitution.

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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