Which of the following lines intersects a circle of radius 5 centered at the origin at two points?

Which of the following lines intersects a circle of radius 5 centered at the origin at two points?

• A. y = -x + 1
• B. y = 2x + 1
• C. y = (1/2)x - 6
• D. y = x + 5

Answer: (B)

To determine which line intersects the circle of radius 5 centered at the origin at two points, we can start with the equation of the circle: \[ x^2 + y^2 = 25 \] A line can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To find the intersection points, we can substitute the equation of the line into the equation of the circle. This will yield a quadratic equation in \( x \). For the line given in option B, \( y = 2x + 1 \), substituting this into the circle's equation will look like this: 1. Substitute \( y \) in the circle equation: \[ x^2 + (2x + 1)^2 = 25 \] 2. Expand and simplify: \[ x^2 + (4x^2 + 4x + 1) = 25 \] \[ 5x^2 + 4x + 1 - 25 = 0 \] \[ 5x^2 + 4x - 24 = 0 \] 3

To determine which line intersects the circle of radius 5 centered at the origin at two points, we can start with the equation of the circle:

[ x^2 + y^2 = 25 ]

A line can be expressed in the form ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. To find the intersection points, we can substitute the equation of the line into the equation of the circle. This will yield a quadratic equation in ( x ).

For the line given in option B, ( y = 2x + 1 ), substituting this into the circle's equation will look like this:

1. Substitute ( y ) in the circle equation:

[ x^2 + (2x + 1)^2 = 25 ]

2. Expand and simplify:

[ x^2 + (4x^2 + 4x + 1) = 25 ]

[ 5x^2 + 4x + 1 - 25 = 0 ]

[ 5x^2 + 4x - 24 = 0 ]

3