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…

Correct answer: A. 4

To find the points of intersection, substitute the expression for y from the linear equation into the equation for the circle. x^2 + (x - c)^2 = 8 Expand the squared term: x^2 + (x^2 - 2cx + c^2) = 8 Combine like terms and rearrange into the standard quadratic form ax^2 + bx + c = 0: 2x^2 - 2cx + (c^2 - 8) = 0 For the system to have exactly one solution, this quadratic equation must have exactly one solution. This occurs when the discriminant (b^2 - 4ac) is equal to 0. In this equation, a = 2, b = -2c, and c = (c^2 - 8). Set the discriminant to 0: (-2c)^2 - 4(2)(c^2 - 8) = 0 Simplify the equation: 4c^2 - 8(c^2 - 8) = 0 4c^2 - 8c^2 + 64 = 0 -4c^2 + 64 = 0 64 = 4c^2 16 = c^2 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 (r^2) in the circle's equation. C is incorrect because 16 is the value of c^2, not c. D is incorrect because 2*sqrt(2) is the value of the radius (r), which would result from an incomplete or incorrect application of the solution method.