Final answer:
The question seeks the slope (m) of a line tangent to two intersecting circles at P(6,4), where both circles are tangent to the x-axis, and the product of the radii is given as 52/3.
Step-by-step explanation:
The question revolves around finding the slope (m) of a line that is tangent to two circles intersecting at point P(6,4). Both circles are also tangent to the x-axis. Given that the product of the radii of the two circles is 52/3, we are to determine the value of m.
To find the slope, we can visualize the scenario and use the properties of tangents and right-angled triangles formed by the radii and the tangents at the points where the circles are tangent to both the x-axis and the line y=mx.
Since the circles intersect the x-axis and are tangent to it, the radii to those points of tangency are vertical and thus are equal to the y-coordinates of the centers of the circles.
Because the tangent line at any point of a circle is perpendicular to the radius at that point, the tangent line y=mx will form a right angle with the radius.
To determine the slope, we must consider the right triangle formed by the radius as one leg and the portion of the tangent line within the triangle as the other leg. We can use the point P(6,4), the x-coordinate of the center of a circle (since it must be on the x-axis), and the given product of the radii to find the slope m.