how do i determine the point of intersection of a circle with a radius of 14 and the line coming in at a 35 degree angle

depends on the line. You have the slope of the line, but no idea where it is. Maybe it doesn't even intersect the circle.

Now, if the line goes through the center of the circle, and the center of the circle is at (0,0) then they intersect at

(14cos35°,14sin35°)

Naturally, if the center is at (h,k) then they intersect at

(h+14cos35°,k+14sin35°)

Also, they intersect at a point diagonally opposite from the above. If the line does not go through the center, then all bets are off.

To determine the point of intersection between a circle with a radius of 14 and a line coming in at a 35-degree angle, you can follow these steps:

1. Find the slope of the line: The slope of the line can be found using the angle it makes with the x-axis. In this case, the 35-degree angle is measured counterclockwise from the positive x-axis. The slope can be determined using the formula: slope = tan(angle).

slope = tan(35 degrees)

2. Determine the equation of the line: Using the slope-intercept form, where the equation of a line is represented as y = mx + b (m being the slope and b being the y-intercept), you need to find the y-intercept (b). To do this, you may use one of the given points or the equation of the circle. Since the circle equation is not given, let's assume the line passes through the origin (0,0).

Since the line passes through (0,0), we can substitute x = 0 and y = 0 into the equation:

0 = slope * 0 + b

As a result, b = 0.

Therefore, the equation of the line is: y = slope * x.

3. Substitute the equation of the line into the equation of the circle: Substitute y = slope * x into the circle equation. The circle equation is represented as (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

(x - h)^2 + ((slope * x) - k)^2 = r^2.

4. Solve the quadratic equation: The equation obtained in step 3 is quadratic. Rearrange it to get it into standard form (Ax^2 + Bx + C = 0) and solve for x. You can either simplify and solve manually or use numerical methods or software to find the solutions.

5. Substitute the values of x obtained in step 4 back into the line equation (y = slope * x) to find the corresponding y-coordinates.

Once you solve the quadratic equation and find the x and y values, you will have the coordinates of the intersection points between the circle and the line.