Answer:
the area of triangle OAP is 12 square units.
Explanation:
First, we need to find the coordinates of the center of the circle, which is (0, 0), and the radius, which is √40 = 2√10.
Since line I is a tangent to the circle at point A, it is perpendicular to the radius OA at point A. Therefore, OA is perpendicular to line I and forms a right angle with it.
We can find the equation of line I using the point-slope form:
slope of radius OA = (0 - 6) / (0 - 2) = -3
slope of line I = 1 / 3 (because it is perpendicular to OA)
equation of line I: y - 6 = (1/3)(x - 2)
simplifying, we get: y = (1/3)x + 4
To find the x-coordinate of point P, we need to solve for x when y = 0:
0 = (1/3)x + 4
x = -12
Therefore, point P has coordinates (-12, 0).
Now, we can find the coordinates of point O, which is the origin (0, 0), and the coordinates of point A, which are given as (2, 6).
To find the area of triangle OAP, we can use the formula for the area of a triangle:
Area = (1/2) x base x height
We know that OA is the base of the triangle and its length is the radius of the circle, which is 2√10.
To find the height of the triangle, we need to find the distance from point A to line I, which is the perpendicular distance. We can use the formula for the distance between a point and a line:
distance = |ax + by + c| / √(a² + b²)
where a, b, and c are the coefficients of the equation of the line, and x and y are the coordinates of the point.
In this case, the equation of line I is y = (1/3)x + 4, so a = -1, b = 3, and c = -12.
Substituting the values, we get:
distance = |(-1)(2) + (3)(6) - 12| / √((-1)² + 3²)
distance = |6| / √10
distance = 3√10 / 5
Now we can substitute the values into the formula for the area of the triangle:
Area = (1/2) x base x height
Area = (1/2) x 2√10 x (3√10 / 5)
Area = 6/5 x 10
Area = 12
Therefore, the area of triangle OAP is 12 square units.