The equations above represent a circle and a line that intersects the circle across its diameter. What is the point of intersection of the two equations that lies in Quadrant II ? A) \((-3 \sqrt{2}, 3 \sqrt{2})\) B) \((-4,2)\) C) \((2+\sqrt{3}, 2)\) D) \((2-3 \sqrt{2}, 3 \sqrt{2})\)

Let's identify the given equations from the problem statement. We are given a circle equation and a line equation that intersects the circle across its diameter. Circle equation(1): \(x^2+y^2=r^2\) Line equation(2): \(y=ax+b\) Here, r is the radius of the circle, a is the slope of the line and b is the line's y-intercept.

In order to find the points of intersection of the circle and the line, we'll substitute the line equation (2) into the circle equation (1). \(x^2 + (ax + b)^2 = r^2\)

Now we'll simplify the equation we obtained in Step 2 and solve it for x. \(x^2 + a^2x^2 + 2abx + b^2 = r^2\) This is a quadratic equation in x, so we'll use the quadratic formula to solve it for x. \(x = \frac{-2ab \pm \sqrt{(2ab)^2 - 4(a^2 + 1)(b^2 - r^2)}}{2(a^2 + 1)}\)

Once we have x, we can replace it in the line equation (2) to get the corresponding y-coordinates of the points of intersection. \(y = ax + b = a \cdot \frac{-2ab \pm \sqrt{(2ab)^2 - 4(a^2 + 1)(b^2 - r^2)}}{2(a^2 + 1)} + b\)

Now we have both x and y-coordinates of the points of intersection. We'll look at each option (A, B, C and D) and see which one lies in Quadrant II. Quadrant II has negative x values and positive y values, so we can eliminate any option that doesn't meet these criteria. A) \((-3 \sqrt{2}, 3 \sqrt{2})\) - This fits the criteria (negative x, positive y) B) \((-4,2)\) - This fits the criteria (negative x, positive y) C) \((2+\sqrt{3}, 2)\) - Does not fit the criteria (positive x, positive y) D) \((2-3 \sqrt{2}, 3 \sqrt{2})\) - Does not fit the criteria (positive x, positive y) Now we have two options left: A) and B). As the given equations are not explicit (line and circle equations are given in general form), the exact intersection point cannot be determined with the given information. However, one of the options, A) or B), should be the correct one based on the specific problem's given information. Considering the given options, we can assume: The point of intersection of the two equations that lies in Quadrant II is: Option B) \((-4,2)\)