Final answer:
To find the area of triangle AOB, we first need to find the coordinates of points A and B and the length of AB. Then we find the equation of the line perpendicular to AB passing through point P to find the height of the triangle. Finally, we use the formula for the area of a triangle to calculate the area in terms of t.
Step-by-step explanation:
To find the area of triangle AOB, we first need to find the coordinates of points A and B. Given that the line has a gradient of -2 and passes through the point P(3t,2t), we can use the slope-intercept form of a line to find the equation of the line. The equation of the line is y = -2x + 2t. To find point A where the line intersects the x-axis, we set y = 0 and solve for x. Substituting y = 0 into the equation, we get -2x + 2t = 0, which gives x = t. So point A is (t, 0). To find point B where the line intersects the y-axis, we set x = 0 and solve for y. Substituting x = 0 into the equation, we get y = 2t. So point B is (0, 2t).
Now we can find the length of AB, which is the base of the triangle. Using the distance formula, we have AB = sqrt((t - 0)^2 + (0 - 2t)^2) = sqrt(t^2 + 4t^2) = sqrt(5t^2).
To find the height of the triangle, we need to find the equation of the line perpendicular to AB passing through point P(3t, 2t). The slope of the perpendicular line is the negative reciprocal of the slope of AB, which is 1/2. Using the point-slope form of a line, the equation of the perpendicular line is y - 2t = 1/2(x - 3t), which simplifies to y = 1/2x - 5t/2.
To find point C where the perpendicular line intersects the x-axis, we set y = 0 and solve for x. Substituting y = 0 into the equation, we get 0 = 1/2x - 5t/2, which gives x = 5t. So point C is (5t, 0).
Now we can find the height of the triangle, which is the distance from point P to the x-axis. Using the distance formula, we have PC = sqrt((3t - 5t)^2 + (2t - 0)^2) = sqrt(4t^2 + 4t^2) = sqrt(8t^2) = 2t sqrt(2).
Finally, we can find the area of triangle AOB using the formula for the area of a triangle, which is 1/2 * base * height = 1/2 * sqrt(5t^2) * 2t sqrt(2) = sqrt(10)t^2.