Final answer:
To calculate the area of triangle OAP, we need to find the coordinates of point P where the line crosses the x-axis. Then, we can use the formula for the area of a triangle to find the area of triangle OAP. The area is 3.5√50 square units.
Step-by-step explanation:
To calculate the area of triangle OAP, we need to find the coordinates of point P. Since line l is tangent to the circle, it is perpendicular to the radius at point A. Therefore, the slope of line l is equal to the negative reciprocal of the slope of the radius. The slope of the radius is given by the derivative of the equation of the circle at point A, which is -1/7. So the slope of line l is 7. With a known point (1,7) and a slope of 7, we can find the equation of the line as y = 7x.
Next, we need to find the x-coordinate of point P, where the line crosses the x-axis. Since the y-coordinate of point P is 0 (on the x-axis), we can substitute y = 0 into the equation of the line and solve for x. So 0 = 7x, which gives x = 0.
Now that we know the coordinates of points O (0, 0), A (1, 7), and P (0, 0), we can use the formula for the area of a triangle, which is (base * height) / 2. The base of the triangle is OA, which has a length of √(1 - 0)² + (7 - 0)² = √1 + 49 = √50. The height of the triangle is the length of the perpendicular from point P to line OA, which is the distance from P to the origin. This distance is √(0 - 0)² + (0 - 7)² = √0 + 49 = √49 = 7. Therefore, the area of triangle OAP is (√50 * 7) / 2 = 7√50 / 2 = 3.5√50 square units.