In the xy-plane, a circle with radius r has its center at (0, k), where k > r > 0. A line that passes through the origi…

Let O be the origin (0, 0) and C be the center of the circle (0, k). Because the line OP is tangent to the circle at point P, the radius CP is perpendicular to the tangent line OP, forming a right triangle OPC with the right angle at P.

The hypotenuse of right triangle OPC is the segment OC, which lies along the y-axis and has length k. The leg CP is a radius of the circle, so CP = r. By the Pythagorean theorem, the length of the tangent segment OP is √(OC² - CP²) = √(k² - r²).

The x-coordinate of point P is the perpendicular distance from P to the y-axis. Let Q be the projection of P onto the y-axis, so that segment PQ is horizontal and perpendicular to OC. In right triangle OPC, segment PQ is the altitude to the hypotenuse OC. The area of triangle OPC can be expressed in two ways:

Area = (1/2) · OP · CP = (1/2) · OC · PQ

Substituting the known lengths:

(1/2) · √(k² - r²) · r = (1/2) · k * PQ

PQ = (r · √(k² - r²)) / k

Because the tangent line has a positive slope and the center is on the positive y-axis, P lies in Quadrant I, so its x-coordinate is positive and equals the length PQ.

Therefore, the x-coordinate of point P is (r · √(k² - r²)) / k.

Choice A is incorrect because √(k² - r²) is the length of segment OP from the origin to P, not the horizontal distance from the y-axis to P.

Choice C is incorrect because (k² - r²) / k is the y-coordinate of point P, obtained from the projection OQ = OP² / OC.

Choice D is incorrect because r² / k represents the vertical distance CQ from the center C to the projection of P on the y-axis, rather than the horizontal distance PQ.