In the standard (x, y) coordinate plane, a circular running track is modeled by the equati… — ACT

In the standard $(x, y)$ coordinate plane, a circular running track is modeled by the equation $(x - 2)^2 + (y - 3)^2 = 25$ . A straight pathway, modeled by the line $y = 2x - 1$ , cuts through the track. What is the distance, in coordinate units, between the two points where the pathway intersects the track?

A
$5$
B
$2\sqrt{5}$
$10$ Answer
D
$20$
E
$50$

Answer

The distance between the two intersection points is

10

The circle

(x - 2)^2 + (y - 3)^2 = 25

has a center of

(2, 3)

and a radius of

r = 5

. Since the line

y = 2x - 1

passes through

(2, 3)

, the line contains a diameter of the circle. The distance between the two intersection points is therefore the diameter of the circle, which is

2r = 10

Step-by-Step Solution

Identify the center and radius of the circle from its equation

(x - 2)^2 + (y - 3)^2 = 25

The center is

(2, 3)

and the radius is

r = \sqrt{25} = 5

To understand the geometry of the circle and find its radius.

Determine if the line

y = 2x - 1

passes through the center of the circle

(2, 3)

by substituting the coordinates into the equation.

3 = 2(2) - 1 \implies 3 = 3

, which is true.

If the line passes through the center, the distance between the intersection points is simply the diameter of the circle.

Calculate the diameter of the circle.

\text{Diameter} = 2r = 2(5) = 10

The distance between two opposite points on a circle passing through the center is equal to the diameter.

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Systems of Linear and Non-Linear Equations

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