y = x - c x^2 + y^2 = 8 In the xy-plane, a line with equation y = x - c and a circle with equation x^2 + y^2 = 8 have e…

To find the points of intersection, substitute the expression for y from the linear equation into the equation for the circle.

x² + (x - c)² = 8

Expand the squared term:

x² + (x² - 2cx + c²) = 8

Combine like terms and rearrange into the standard quadratic form ax² + bx + c = 0:

2x² - 2cx + (c² - 8) = 0

For the system to have exactly one solution, this quadratic equation must have exactly one solution. This occurs when the discriminant (b² - 4ac) is equal to 0.

In this equation, a = 2, b = -2c, and c = (c² - 8).

Set the discriminant to 0:

(-2c)² - 4(2)(c² - 8) = 0

Simplify the equation:

4c² - 8(c² - 8) = 0

4c² - 8c² + 64 = 0

-4c² + 64 = 0

64 = 4c²

16 = c²

Take the square root of both sides to find the possible values for c:

c = 4 or c = -4

The question asks for one possible value of c. Of the choices provided, 4 is a possible value.

B is incorrect because 8 is the value of the radius squared (r²) in the circle's equation. C is incorrect because 16 is the value of c², not c. D is incorrect because 2√(2) is the value of the radius (r), which would result from an incomplete or incorrect application of the solution method.