Question

Nellie is analyzing a circle, y² + x² = 25, and a linear function g(x). Will they intersect?

a) Yes, at a positive x-coordinate or zero.

b) Yes, at a negative x-coordinate or zero.

c) Yes, at a negative and positive x-coordinates or zero.

d) No, they will not intersect.

Answer 1

To determine if a circle and a linear function intersect, substitute the function into the equation of the circle and solve for x. If the discriminant is greater than or equal to zero, they intersect; otherwise, they don't. The answer is d) No, they will not intersect.

Step-by-step explanation:

The equation y² + x² = 25 represents a circle with center at the origin (0,0) and a radius of 5. The linear function g(x) can be represented as a straight line. To determine if they intersect, we can substitute g(x) into the equation of the circle and solve for x. If the resulting equation has solutions, then they intersect. Let's solve:

Substitute g(x) into the equation: (g(x))² + x² = 25

Expand and simplify: (mx + b)² + x² = 25

Expand further and combine like terms: m²x² + 2mbx + b² + x² = 25

Rearrange the equation: (m² + 1)x² + (2mb)x + (b² - 25) = 0

This is a quadratic equation in x. If the discriminant (the expression inside the square root) is greater than or equal to zero, then there are real solutions for x and the circle intersects the line. Otherwise, there are no intersections.

Therefore, the answer is d) No, they will not intersect.