Question

(6 marks)

1. The circle C has equation x² + y² - 10x + 8y + 25 = 0. The line with equation y = kx, where k
is a constant, cuts C at two distinct points. Find the range of possible values for k.

Answer 1

The range of possible values for k is k < -3 or k > 5/3.

Step-by-step explanation:

To find the range of possible values for k, we need to solve the system of equations formed by the circle equation and the line equation.

First, we substitute the value of y from the line equation into the circle equation:

x² + (kx)² - 10x + 8(kx) + 25 = 0

Simplifying this equation, we get:

(k² + 1) x² + (8k - 10) x + 25 = 0

This is a quadratic equation in x. For the line to cut C at two distinct points, the quadratic equation must have two different real solutions.

Therefore, the discriminant of the quadratic equation must be greater than zero:

(8k - 10)² - 4(k² + 1)(25) > 0

Solving this inequality, we find:

k < -3 or k > 5/3

Therefore, the range of possible values for k is k < -3 or k > 5/3.