Question

The line r, of equation 2x + y - 1 = 0, and the circumference of equation x² + y² + 6x - 8y = 0, are given. What is the position of the line r in relation to the circumference?

A. The line r is tangent to the circumference.

B. The line r intersects the circumference at two points.

C. The line r does not intersect the circumference.

D. The position cannot be determined from the information given.

Answer 1

The line r, with equation 2x + y - 1 = 0, intersects the circumference, with equation x² + y² + 6x - 8y = 0, at two points.

Step-by-step explanation:

The line r, with equation 2x + y - 1 = 0, and the circumference, with equation x² + y² + 6x - 8y = 0, are given. To determine the position of the line r in relation to the circumference, we need to solve the system of equations.

1. First, solve the equation 2x + y - 1 = 0 for y:

y = -2x + 1

1. Substitute this expression for y into the equation x² + y² + 6x - 8y = 0:

x² + (-2x + 1)² + 6x - 8(-2x + 1) = 0

x² + 4x² - 4x + 1 + 6x + 16x - 8 = 0

5x² + 16x - 7 = 0

1. Solve this quadratic equation for x using the quadratic formula:

x = (-16 ± √(16² - 4(5)(-7))) / (2(5))

x = (-16 ± √(256 + 140)) / 10

x = (-16 ± √396) / 10

x = (-16 ± 2√99) / 10

x = (-8 ± √99) / 5

1. Substitute these values of x back into the equation 2x + y - 1 = 0 to find the corresponding values of y:

For x = (-8 + √99) / 5, y = -2((-8 + √99) / 5) + 1

For x = (-8 - √99) / 5, y = -2((-8 - √99) / 5) + 1

1. Thus, the line r intersects the circumference at two points.