Final answer:
The student is tasked to calculate the slopes of the tangent line y=mx+c at the point (8,3) to the given circle's equation. By utilizing the derivative to find the tangent at the point of contact and solving the system of equations, the slopes can be determined.
Step-by-step explanation:
The question asks to find the two possible values of the gradient (slope) of the line y=mx+c that passes through the point (8,3) and is tangential to the circle with the equation x^2+y^2-4x-6y=-4. To find the gradient of the line, we must use the fact that the line is tangent to the circle. This means that at the point of tangency, the derivative of the circle's implicit equation will yield the slope of the tangent line. Since we have a specific point (8,3), we can substitute into the line's equation to determine the value of c. After finding c, we will have two equations representing the line and circle, which we can solve as a system of equations to find the slope(s) at the tangency point(s).